Question for those who believe to their irrationality of Pi demonstrations.

- algebraic + proof + irrationality + of + pi (USA California Santa Cruz Univ.)

Any algebraic proof of irrationality is done according to the mathematic convergence methods and to successive Euclidean divisions. These types of methods are necessarily the unbounded accesses methods denoted as irrationallity. Therefore, the irrationality is not concerning a numbers characteristic but only the irrationality of bad mathematic's methods. A bad method is only a false method and can never be classified as a mathematic irrationality discovering.


- formulas + N + "which" + pi + appears + in (USA)

Actually, we have the natural Integer numbers series that is written as :
N = Pi.x
as shows in the following where N=circumferences and x=diameters.

- irrational habit plane (Shiraz University, Iran)


The proof of circle surface (S) in the following uses summation of circle perimeters or circumferences (P) that compose it. Then, one get (P) versus diameter (D) :
P = Pi.D
This perimeter is then a linear function of the diameters.
Consequently, any circle perimeter is function of diameters varying from zero to any number of the universe. Then, this perimeter (P) can vary from zero to any number of the universe. Particularly all these (P) numbers necessary contains the natural Integer series beginning from 1 to unbounded great quantities. The figure here, shows the graph of this linear function.

Thus, why a product of the irrational Pi give raise to the natural integer number series ?

Every provers of the irrationnality of Pi, have to make a point of giving an answer about a such FACT : why irrationality produces rationality ?


- irrational + numbers (Niger)

Here is only a particular exemple. There is a basic proof that irrational numbers are contradictions. Because it can be proved that any Natural Number is irrational with an exhaustion method. And inversely, we can prove with compass and edge that any number, include the whole Natural series ones, that all quantities can be precisely and exactly determined.

Theorem on integral calculus for a circle aera

- PI + "MATEMATICS" (Slovenia)

... a new most simple but exact method. - mathématics + integral + concepts + and + formule (Mexico)

- how + many + degres + has + a + circle (Canada)
AAny given value is valid as representing a circular cycle. Therefore, one (1) is valid and any natural integer number is valid. It is why the space constant (pi/4) is rational because an operation on an irrational number is known as producing irrational result. This prove also that irrational mathematics are the most confude and contradictory formalism.



- why + are + circle + graphs + mostly + "alway" + named + circle + O ? (USA)

... HELP !!!


- physic + and + technologies + invention + 2002 (Phillipines)

- the + universe + observes + itself (USA)

Because matter is memory and the thinking is constructing reason on memory data the universe seems observing itself. However, this matter process is not a closed loop system. A closed loop produces bounded process from there one can say that the universe is an observed observer. While building New reasons from memory is a Succession process. Because this latter is a Recurrent one, the alive universe is unbounded existence based on a finite quantity of matter. From there it appears as Matter Conservation principle conserbation. Therefore, the universe of the alive is an open logic system based on permanent foundation.

- law + of + "conserbation" definition (Canada)

Finger motions is not an uniform one... this the last experimental statement.



- pi + value (Hungary)

- proof + that + pi + is + transcendent (Israel)

- determination+of+the+value+of+pi (Ghana)

- counted + pi + value (Finland)

- taoism + and + concentric + circles (USA)

- the+universe + observes + itself (USA)

- What + methods + have + been + used + to + find + a n + accurate + value + for + pi (Montserrat)

- finding + pi + by + rolling + measuring + a + circle

- basics + construction + fondations (Canada)

circles and curvatures are internal properties of circular structure. Any measurements of circles through linear function must necessary uses the Constant factor of spatial circle transformation. In the following, is a theorem on circular aera integral calculus measurements using circumference in discrete matter :

- what + is + Circularity + maths + and + science (Zambia)

- pi + value (Maldive)

- parallel + world + equations + the + proof + that + "ther" + is (Australia)

- why + circle + of + life + and + not + a + square (South Africa)

- quantity + of + cars (China)



- transformation + from + square + to + circle (USA)

- MATHEMATICIANS + CIRCLING + THE + SQUARE (Australia)

- mécanique + propriété + matière (Vietnam)

- mathematic + formula + to + calculate + circle + lenght (Latvia)

- the + meaning + of + the + "ying" + and + the + yang (Ivory Coast)

Actually, beginning of life was born from two matter structures that contribute all along the universe history to furnish basic materials and reasons developments for construction. As a Mother and a Father are two complementary structures linked together to be able to prepare a child, they are characterized as the yin and yang principle of life. The two necessary circle and squarre structures with which an unbounded way was opened. The Yin and Yang are not ideology but only observations of the reality of the universal life possibilities extending on heavenly globe bodies that fit to physic conditions.


- square + and + round + archetypes (USA)

CIRCUMFERENCE PREAMBLE

- spatial figures (Philippines)

You are right to underline such notation.
Denoting of "spatial figures" is specific to the Dakhiometry language to define spatial constructions of figures using basic elements of space as point, line and curve.
Obviously one use to tell that it is Geometry. However, it should be distinguishe the Dakhiometry from the old Geometry. Even if the forms seems to be the same but the mode of thinking is quite different because the spatial language rests on true principes of the universe's concrete and discrete absolute space. In fact Geometry is only figure formalism abstraction and ignores what is the universe space. For Geometry, the space is rather the inheritance of the Anciant Nothingness ideology. It is why Geometry was early a dead-language and was overcome by Algebra and Anlaytic Geometry methods that use Number formalism jointly to graphics. These latter are not spatial constructions but resulting from numeric algoritthms.
The spatial language is unbounded in fertil manners. For example calculation are done by simple easy and fast construction using space elements. It is done in precision and exactness if point location without suffering errors from neither irrational Number holes nor infinity limits of "not-a-number" methods.

- Day length as a function of latitude (USA, Mary Land)

- constructing spatial figures (Philippines)

- varying length of day and night (Philipinnes)


Length of days along the Earth ecliptic can be defined by spatial figure as the next one.
On this figure, the North altitude is shown from 0 to 90 degrees. Length of days are defined by different latitude circle perimeters. All these circle denote 24H time of a day. They are face to the Sun normal view toward the Earth but on this figure is represented what does the Sun see on different Earth globe latitudes. The red part of any circle are durations of a days according to its latitude and the reminder of its perimeter in blue color denotes the Night duration.
This figure represents the Winter and Summer Earth solstices for the North hemispher. For the South one, we have to permute the color of the perimeter as night for day duration and inversely.

Fot the equinox in Spring and Autumn the Earth pole axis is in a plane perpendicular to the Sun radius. This produce an equi-axis Earth distance from the Sun. Therefore, the 24H circle duration is divided in two equal parts for the day and night perimeter lengths.




- convert a circle to a rectangle (USA nih gov)

- squaring + the + circle + equation (UK)

- squaring + the + circle (Belarus)

- impossible algebra formula (USA New Jersey)

- four bugs simultaneously crawling towards each other, square, physics (USA NMT edu)

- Sacred number formula revealed


- description of E.T.'s over time (USA)

- sacred physics (King Country Library System, USA)

- circle pi circumference (microsoft.com, USA)

- optimal disposition of squares (Canada)

- proof + squaring + a + circle (The University of Texas at Austin, USA)

- proof with mathematical pi (american.edu)

- the root of all action square root of two (bell.ca, Toronto, Canada)

You have get an insight of action practice. So, plough more further to gather in the crops.

- nuumber pi figure (charter.com, Missouri, USA)







Remind


Resume to this figure:


Let S_AD the greater square surface of side [AD] et S_AB the one of the little square of side [AB]. The ratio of these surfaces can be written as :
S_AB/S_AD = Pi/4

This a form of the Space Constant given from the Squaring a circle. It will be very usefull in the following.


- "STARIGHT" LINES FORMULA (Malaysia)

       y = (------------------>)

- The circle perimeter formula.

The universe Existence is as a toggle of light. But at the beginning light was not a phenomenon. The universe beginning is rather a toggle of knowledge owing to the starting possibility of transformation of curvature into straignt straight line. We human can know this universe event using the space structure logic. Then performing a squaring the circle we can get the process of how space use to go from a curve into a staright line. The square was a new space structure born from the circular one where measurment can proceed for the beginning of the universe construction.

We have seen that the squaring a circle is done using a space Constant. Then we can know how to transform any circular line into a straight one and inversely.

This transformation also must be refered to the basics space structure where the universe is originated. It is the circle and the square associated structure. This latter is a circle inscribed in a square or inverseley a circle circonscribed to a square. This two forms are related between them by the space Constant. Therefore, we can write the immutable relation between the perimeter of a square and the circumference. If (D) is the side of the square and the circle diameter; (P) the perimeter of the square; and (C) the circumference, we have :
P = 4.D
Then,
C = P . pi/4
Inversely,
P = C . 4/pi

The space Constant is used here as (pi/4) or its inverse (4/pi).

According to these relationships, the circumference formula is given as :

C = pi.D It corresponds to,

C = P.(pi/4)

- fundamental + theorem + proof + different + equation (Vietnam)

- Reciprocal + formula + for + construction (Hongkong)

- when + was + the + first + circle + invented? (Former Czecholovakia)

- formula + for + finding + the + perimeter + of + a + square + when + you + know + the + "diaginal" (USA)

- circle + perimeter + formula (Singapore)

Bows practical application

The space Constant is then means of figure out any circular bow lengths. Inversely any straight length can be define a bow length legth in a any circle. This can be done in any dimension dimansion. These applications should use the corresponding suqare square-cercle structure. Example is show in this next figure in that there are drawn two equal circle bows and their corresponding half square straigth lengths. In theses structures an angle define a proportion of the bow and it equivalent same proportion of length, in the square perimeter. Basic square-circle structure are square and circle inscribed or circonscribed. From these equivalent, one can extend this equivalence to any dimension according to proportion defined.

- "legth" + circular + loop (USA)

- finding + "suqare" + root + algebraically (India)


- tiny + pi + calculation (Italia)

- bow + distribution + exercices (USA)

- surface + "area cylindar" (USA)

Why the empirical process of convergence
by exhaustion like the polygon one,
is an error erreor when founded as an axiom

- manipulate + nothingness (Germany)

A fine meaning in words association.


- trancendent + classes (Mexico)

- define + Tchuan (Canada)

- polygone + inscrit + et + circonscrit (Canada)

We have now the mean of the transformation process between straight line and curves. It allows to understand why the Archimedes' exhaustion can't be a success and why with modern calculation machines polygon exhaustion can't get the excat Pi value.
The polygon axhaustion method is a very fine idea and eveyone of the best may end to use this polygon method because it seems the most efficient.

One don't have to mask this event and reminds that no mathematicians from Archimedes to all the modern ones, nobody never being able underline some controversial point about the old pi value. The Squaring a circle was forbiden as impossible problem and also as heretic subject. The more, since during four years, my publication on this site about the squaring a circle meet silence from any scientific institution over the world. This last fact is a real aberrant heretic occurence that is a real odd thing today. So, for all institutional matheamticians, I hope you will fully take advantage and make the most opportunity of the Squaring a circle fertilities an also to avoid so many errors generated from the current methods.

This is only a particular point of the Dakhiometry. This latter opens up more larger new horizon that mathematics can't never predict.

- pi + structure (Cocos Island)

It is the just concept : a space structure constant. There is large consequences that simplify the space knowledge.


- answers + for + "Archemides" + problems + in + high + school (Germany)

- Archemides + organization + in + estonia (regionaalhaigla.ee, North Estonia Medical Centre, Regionaal haigla, Estonia)


- define + physical + factors (Jamaica)

- squaring + or + "squarring" (USA)

- ultimate value of PI (USA)

- algebraically + prove + that + a + circle + has + a + larger + "area" + than + a + polygon (UK)

- how + many + segments ++ will + fit + on + "teh" + circumference + of + this + circle (USA)
There is no number of linear segments that can correspond one-to-one to a circonference, even if we can count the immutable elemental matters. Because linear and curvilinear lines are of two quite different logic worlds. When operating in the linear world, and try to resolved circle as a polygone constituted by points as sides, comes from ignorance of this basic space difference. The only means of comparing these two worlds is not to directely linked them point by point, but to leave the one and enter into the othern by means of the Space Constant.
Ignoring the Space Constant, drives Galileo to prove that an infinite line correspond one-to-one with a half-cirle length. This is the infinitesimal Continuum world where every thing can be confused and allows to produce many dogma.



- CIRCUMFERENCE + OF + A + circle + USING + LINE + INTEGRALS (UK)

- discovering + pi + as + the + limit + of + perimeters + of + polygons (Canada)

Convergence and Exhaustion methods errors.
Ignorance of matter space

- draw square with compass (USA, Texas, Abilene McMurry University)

- périmètre + polygone + exinscrit + supérieur + cercle (France, Lyc-Curie-Sète Ac-Montpellier)

On the Circle or Mathematics an Physics ignorance of space

About Plato theories
- Plato's inventions (USA)

- the world of form (Thailand)

May be, from the Plato's compass was intituted the famous circle definition. From there circle formula are defined by a radius. It is only a creation from "idea".
Because practically and more basically, a circle should be accessed by its diameter. Remember that manufacturing of tubes use a diameter to specified any tube. And the first possible measurment of circular form is the diameter. Also, astronomy uses diameter of heavenly bodies.


Property of space in Dakhiometry
A circle can't be construct with nothing else than according to the circular properties.

Plato tell that the world is based on the differents geometrical figures included the circle. But this universe space description shows that he don't know really what is the circular realm because a such theory is only the Linear universe.

Because when he tell that geometry should be constructed with a compass and a straight edge he din't be in consciousness that a compass is not a circular device.
a Compass, is only a LINEAR device!
Therefore, the expression "compass and straight edge" is equivalent to "Staright edge and straight edge".

On this next figure, is shown the real basic structure of a compass. A compass is no other thing than a straight segment. It is a Linear structure. The rounds done with a compass are only conventional symbols of Circular Circles produced by a linear tool. While a square is really a real linear figure constructed with linear tools.

Definition of a circle
A circle is a structure of space locations in Circular Succession. Nothing else out of this circular succession belong to a circle property.

Thus, the definition of a circle as points at equal distance from a center point, is only a figure of the Linear Realm. It is why most statements on circle are done using its radius. This is the proof of Mathematics ignorance about the basic difference between Linear and Circular properties.

Consequences
a) - According to these usual errors about a compass, one can draw many geometrical figures with a compass ignoring that one is using only a straight edge structure and the circle are there only a conventional figure of "equal distance from a center point".
b) - In practice a square can't be constructed using only with the property of a circle as circular points in succession. Because this circular structure is in fact the Direction Structure or more wellknonw tell as Angular Structure.
c) - From a circle there is uniquely one possible transformation: the squaring a circle. This latter is the even from there Linear realm can be revealed. It is the event of linear Direction that allows to measure the Angle in space. Angle can be defined only when straight line exists in conscience.

For example, to draw a square with circle, we need first to define at least a 90° angle or its multiple. We do that because, we KNOW yet what is a direction lines. But with only the real circular structure we can't know what is a straight line.



From Ignorance of the Circular realm,
appears the empirical mathematic of Exhaustion and Convergence methods
Ignoring that space is characterized in two aspects Linear ans Circular, one meets difficulties of knowing what is a circle.

There is the famous problem of defining the circular structure as for example the circle diameter and its disk aera. A mysterious and magic key of transformation appears in the whole world as the unattainable God with the name Pi.

From a such God, Mathematicians are running in a vicious circle as the Achiles and the Tortoise race.
For this purpose, theories are built for the know how to catch a ghost as follows:

a) - Zeno's Achiles' axiom
Running after, you can't catch nothing if you are enough doing the half of the necessarry way to be equal the job.

b) - Archimadean enhance axiom
To catch an aim you need to do successively more than a half-way toward the goal. You will then going byond it.

- Archimedes' + principle + is + not + precise + enough + for (Turkey)

In fact the preciseless of this axiom is only because it is an empirical statement. Discrete Space is the fullness of precision. When nothing is known one use to think "more than and less than principle". It is also the case for the Ptolemy theorem of inscribed quadrilaterals where the precision can be shown.

According to these famous and applied theories, here is how Archimedes try to catch the shifty God Pi. It is compare there what are the difference between a Half-a-way denoted as (M_Z) versus the more-than-half-a-way denoted as (M_A).

- pictures of plato and his inventions (myvzw.com, MicroSoft Corp, USA)

Actually, the two segments of a nice compass device is only one segment of the circle radius. It is why one use to define a circle with its radius instead of his diameter. This later is what in astronomy pactise we need to know and can be measured directly.

- Plato's belief in law (rogers.com, Canada)

- ignorance exhaustion death (FHCRC.ORG, Fred Hutchinson Cancer Reasearch Center, Seattle, Washington, USA)

- Plato's inventions (California, USA)

In this figure is shown (OB) as what is a Plato's Compass.

- Plato's inventions (optonline.net, Canada)

- Plato theory of forms (sky.com, UK)

- Plato's belief (k12.wi.us, Madison Wisconsin, USA)

- Plato theory of ideas (Latvia)

All object construction of the universe are done exclusively according to the universal laws. Whatever are the multiplicity of forms any construction is then deduced from the universe space structure properties. Human is no more different from any simplest object. In fact nothing is really simplistic because they are perfectly organized by the universal laws. Perfect means here according strictly to the Facts of universal laws. Human capability of thinking is no more than a form property of law construction as well as a falling body or a chemical transformation of any matter even if the thinking seems to be a more complex transformation process. The human thinking is only the fact of a complex transformation mode inherent to any human and is being basically impress in human as his own internal property.
Therefore, human Ideas are not something outside to the human whole construction sytem. Human ideas are facts of internal structure of the alive. For example, the fact is that anyone can intuitively know any elemental form in the nature even if he can't named it. It is because, without no form recognition nobody can't neither behave nor move as he need it in evryday life.
Note
that universal laws means that strictly, nothing is outside of any universal law. It is why there is no Nothingness. Nothing is in the Nothingness according to the Nothingness should define No Universal Law because anything is necessary properties from universal laws.

- Plato forgotten river (comcast.net, Tennessee, USA)

- philosophical theory of forms (USA)

- Plato the divided line summary (asu.edu, Arizona State University, USA)

- theory of forms non material (Missisippi, USA)

However, forms are Quantities the property of concrete matter. The belief that ideas are forms, therefore cunfusing ideas with quantities, drives to Dedekind and Cantor to invent from their imaginative ideas, that infinities are also forms, the numbers ones. The Euclidean Line is also a Nothinness in contradiction when designating the material space line. In brief, since millenaries, Mathematics is manipulating Nothingness and ignores what is the concrete forms of space reality. This is the science Continuum dogma background.

- Plato's belief knowledge chart (aubg.bg, Blagoevgrad, Bulgaria)

- examples of what is the difference between reflection and refraction in math (suddenlink.net, USA)

- math pi examples in middle ages (rr.com, Hawaii, USA)

- What Did Zeno do for math (qwest.net, USA)

- maths example -lateral thinking enhancing vertical thinking (ub.bw, University of Botswana)

- image of difference between open-loop system and close-loop system (Russian Federation, http://controls.engin.umich.edu/wiki/images/a/a0/)

The belief is that Linear tool can reproduce the Curvilinear realm.
Difference between Zeno's Clasical Exhaustion (M_z) and the Enhance Archimedean one (M_A), is that Zeno propose to do exhaustion with the preceding half-step while the Archimedean axiom states that it is done with a little more than a half-step. But both are unsuccessful courses.

Also, the Plato's Compass as a segment-OB Principle.

The Archimedean exhaustion process can't never be achieved. because simply, we can only exhaust a polygon toward a PRECISE GIVEN goal and not toward exactly an UNKNOWN thing as the Circle-to-be-found. That is to tell, the polygon method is to catch the shadow of a ghost.

1) - In Dakhiometrie the Archimedes' axiom as {(n.a)>b} is only a consequence of a real basic principle. It can be only a theorem simply proved.

2) - The way from points A and B can be precisely decomposed and fully "exhausted" with the space structure. Point (A) is precisely equivalent to point (B). But only using the Circular property and not the linear archimedean exhaustion step.
Why is there a such result?
It is because in the space structure, there is the immortal presence of the Space Constant!



- moment + of + inertia + circonference (Ireland)

- what + is + the + definition + of + mathematical + term + absolute + difference (USA)

- points + to + note + when + constructing + concrete + surfaces (Guyana)

- mathematic history model on how formula of "area" of circle was derived (USA)


- Plato geometry (tedata.net, Egypt)

From the biginning of human concept on space there is only linear mode of undestanding space. Even if Plato tel that Geometric problems should be resolved with compass-and-straight-edge, he was telling only linear space meaning. The classical impossibility of resolving the squaring a circle and the angle trisection are example of ignorance of the universe space or the matter space. It is because the angle is a circle property. The origin of space ignorance is to be unconscious that the circle is not a form amongst so many other linear forms. So, it was a Plato's errors to tell that tne universe is an assembly of triangle, squares and polygones as layers in spherical sky.
True knowledge of the universe space should rest on the foundation that the circle is not a form but a logic-of-form. The transcendent logic which we have access to, is the Space Constant of Form Transformation exposed in this present text.

1) - circumference definition
The circumference is composed of the number of discrete elemental matters disposed successively in a same central symmetry, two by two. Thus, a circumference is the "area" of a minimal circle crown.

2) - The summation of all the crown "areas" of a circle is then the total "area" of the circle disk.

3) - The "area" of a circular disk is given as :
Theorem :
The "area" of a circular disk (S) is the summation of all the circumferences of concentric circles included in the "area" given (S).

- concentric merci device (India)

- animated bc calculus (USA, Illinois IMSA)

- unfull circles (ntl.com, bm, UK)

- Archimedes measurment circle high resolution (uoc.gr, University of Crete)



Let an integral calculus process of measuring an "area" (S) of a circle (C).

Let (D) the diameter of elemental matter crown and (dx) the minimal length of an elemental matter.
Then the "area" of circle (C) is the summation of all the crowns of diameter (D+dx). Provide that D varies from zero to CD.



- formule mathématique incorrecte (Canada)

Actually, this following formula is nothing.

S = Sommation [ (D+dx).Pi.dx/2 ]       (aa1)

D varies from zero to CD

The disk of diameter D is decomposed to concentric circles, according to elemental succession (dx). The aera of the disk is then the summation of all these concentric circles. The corect formula is as follows:

Summation all perimeters =
Pi[Summation from zero to D of n_i(dx)]       (001)

This algebric formula is only an approximation of axhaustion method.


Reciprocal
- uniqueness + of + the + reciprocal + proof (USA)


This theorem in direct proof is based on the formation of a circle "area" by circumferences of discrete matters.
The inverse proof is as followss:
Does a circle composed of circumferences as defined in the direct mode ?
The answer is yes.
According to the theorem of circle Chords. This latter proves that a circumference of a circle (C) allows to determine by unique manner any points of its radius as belonging to one cocentric interior circonferneces. Therefore, any "area" of a circle can be decomposed into uniques interior and successive circumferences as proved by the direct theorem. The reciprocal is True.

- different + things + built + on + the + basis + of + knowledge + of + pi + in + ancient + period (United Arab Emirates)

- "differents" + values + for + pi (Sweden)

- remote + control + robot + "construcyion" + set (Thailand)

- does + the + perimeter + of + a + polygon + convert + to + a +limit ?(USA)

An important question that needs a fundamental precise definition.

If matter space is a fact of absolute locations of immutable elemental matters. Then, a line composed of matter elements is necessary composed of one elements aside an another all along.

Then, an ultimate side numbers of a polygon is necessary composed of a total number of elemental side lengths. Thus, a polygone has necessary a limit of number of sides and not of side lengths.

So, what is the difference between the successive elements of a polygon and the one of a curve ? The fundamental differences are:

1) - Polygon

1-a) - A such polygone limit is a set of sides that are structured as a polygon.
1-b) - The perimeter of this polygone is a CONSTRUCtION of elemental lengths.
1-c) - The differents different elemental sides of this polygon are PLACED one after the other. Each side is differently DIRECTED from the next one.
1-d) - The polygon is then a set of associated elemental Straight Lines.

2) - Curve

2-a) - A curve is also composed of a number of elemental matters.
2-b) - The length of this curve is a SUCESSUION of elemental matters. It is the RECURRENT structure of matter.
2-c) - Therefore, a curve is a whole ONE DIRECTION recureence structure. This direction is toward one absolude precise point. It is why a curve is located and defined by its center.


Measurement resulting :
At the ultimate boundary of a polygon and the circle perimeter, we have a seemingly elemental line. However, the problem is : WHERE these two lines should be surimposed as equality of form ?

Therfore, one use to think that polygon-circle equality is directely the consequence of a polygon being-a-line with this line staying directely on the circle perimeter. It is only a sensitive obviousness that don't rest on any reason even if multiplication of polygon side number seems to match. It is the same for the infinity descent method there is no logic reason other than a conventional obviousness. As there is no reason to allow the Archimedes' Exhaustion, this situation proves that there is some artifact in the convergence calculation process. These convergences do not drive towards a presupposed value. The proof is given by the Euclidean division algorithm where there is not a necessary end of convergence. And also, in the following exemple of the integral calculus where one ontains the value 3.1416 for Pi. A precise infinitesimal calculus, don't change there the result. Thus, it is really an systematic severe error due the convergence process.

Only the space structure concept, allows to define the difference between circular and straight line based forms.

In brief:
A polygone is a multi-direction line organisation. One name its perimeter as a polyline construction.
A curve is a whole ONE direction Recurrent structure that allow to define precisely an absolute point center.

A polydon polygon that is a particular space construction can't replace a space structure principle represented by a circular line. It is why the biginning of the universe construction happens from a circle mono-space-structure transmuttation that allows multi-direction constructions. Inversely, a construction as a polygon can't directly get into a basic permanent space structure. This is why the universe construction evolution goes forward on an unbounded way without never return back. Possibility of including circular forms as a tools amongst other forms can only be done through a Spatial Constant factor.

Aside approximate processes, there is no other means to precisely surimpose equality between straight lines and curvatures. This is a basic characteristic of matter space that obeys to principle of absilute absolute locations and non by random placement of matter in a nothingness.

- "polydon" with five sides (USA)

- "absilute" + value (USA)



- Which + of + "teh" + following + values + "of" + "ameasure" + quantity + shows + "teh" + "greates" + resolution (Canada)

Actually, it is done with a very low resolution apparatus that can tell anything...


- pi + human (Estonia)

- why + thus + the + value + of + tangent + function + did + not + touch + the + undefined + value (Philippines)

- vertex + determination + angular + resolution + basic (UK)

A fine rational definition of polygon vertex by the variation of direction measurements.


- discreet + mathematics + methods + of + proof + direct (Qatar)

- a + measurement + that + is + accurate + is + one + that + is + as + exact + as + possible + is + close + to + the + true + value Ccontains + at + least + four + significant + figures Cor + contains + five + decimal + places (USA)

- who + invente + pi + number (Dominican Republic) - value + pi + full (Spain)

- the + "mathamatic" + of + beauty (UK)

- atomic + "bonb" (Japan) - "bonb" + atomic (Morocco) - what + are + the + physic + of + tai + chi (USA)

Proof of the circle "area" formula as Pi.R²

- Holly + Bach + Remedy (UK)

- supernovas (Peru)

- how + does + the + formula + "area" ++ Pi + R + square + evolve + + or + come + about (Canada)

- que + es + un + mumeral (Peru)

- more + knowledge + circle + perimeter (USA)

- Planck's + length + proof (USA)

- definition + "spartial" (Singapore)

- having-a-concept-of-self (HongKong

- continuum + of + reality + escapes

- transcendent + pi (USA)

A Sparial Constant as real meaning of concrete construction beginning of the universe.


- To + what + extent + can + the + -rule + of +law- + help + to + establish + just + law (UK)

- absolute + location + of + k2 (USA)

- last + four + supernova + occurrences (USA)

- "spartial" + figures + geometry (Honduras)

- where + is + the + liberty (Brazil)

- prayer + wheels + erase + energy (USA)

- "discribe" + the + properties + of + gold (USA)

- is + there + any + thing + beyond + the + universe? (New Zealand)

- meaning + of + "spartial" (Kenya)

- exact + value + of + pi (Australia)

- pi + value (Costa Rica)

- physic + transformation (Denmark)

- "spartial" + organisation (Tanzania)

- proof + iterative + formula (Ireland)

- is + the + word + Phenomena + a + mathematical + term ? (USA)

- How + many + percentage + of + "Ligth" + is + absorbed + by + paint ?(Germany)

- What: +++ La + Vérité + est + toujours + l'idée + qui fait le tour + du + Monde +++ means (USA)

- archimdean + property + of + R (UK)

- what + quality + consciuosness + and + habit + process + is + all + about (Philippines)

- what + if + an + equation + had + three + variables (Canada)

- numeral + pi (Antigua and Barbuda)


- démonstration + surface + d'un + cercle (Mexico)



- Archimedes + "area" + of + circle + equal + right + triangle + circumference (USA)


- extending + a + known + perimeter + then + finding + the + new + perimeter& (USA)

Exactly tell. Extending a line, is a same line while extending a circle perimeter is creating a new perimeter. This the characteristic of the circular structure. Consequently the universe can be define with nique circle series.

- how + do + you + find + a + square + of + a + series + of + numbers (USA)

- all + seeiing (Argentina)

- finite + elemnts + analysis ++ calculation (Malaysia)


- law + of + conservation + of + maatter + during + a + physical + and + a + chemical + change

- indestructible + elements (USA)



- instruments + to + find + the + physical + quantities + of + "area" + speed volum time lenght density mass + "etc" + and + more (United Arab Emirates)

Y O U   A R E   R I G H T !  "H I G H T"  R I G H T !!!

- properties+of+the+matter (Chile)


- "spartia" l +summation (Kenya)

- proof + of + "area" + of + circle + at + first + time (Pakistan)


- when + was + the + first + "ewhaust" + system + invented (USA)


- "Formaula" + for + Density, Mass + and + Volume (Greece)

- "formaula" + to + find + image + of + a + point + with + respect + to + a + straight + line (India)


- "itegral" (Brazil)


- reseach on failure in using formulas to measure area (Maldives)

- animated area formulas (Philippines, Mindanao State University, Iligan Inst. of Tech.)

- algebraic formulas (Mauritius)

- exact area of a circle using integral notation (comcast.net, Oregon, USA)

- measure circle "area" (Korea)

- 4d Singapore formula (Singapore)




The above summation formula (aa1) is the proof of how a circle "area" is composed of.

The above process uses ultimate elemental construction to define the circle "area". This present method is quite different from the Algebraic integral one itegral. The next figure figures out this summation process written in equation (aa1). It consists to refere on a measured surface of a square and then to deduce the circle surface according to the Space Constant obtained from the squaring a circle.


- DEDUCTION ORDERS ANIMATION (USA Texas)

After the squaring a circle resolution, a Space Constant T_T is known. This latter is the one and unique key from it all the circle characteristics are given. One can deduce precisely and exactly the aera and the perimeter of circle.

- circle "area" and perimeter (South Africa)

With a circle of diameter D we have:

a) circle Aera = T_T(D²/4)

b) circle Perimeter = T_T(D)

From relation (001), we get:

T_T(D²/4)=R²=T_T[Summation from zero to D of n_i(dx)]

we get:
D²/4 = [Sommation de zéro à D de n_i(dx)]

Let a square of side equal to D. It can be seen that the circle aera is:

Disk Aera = (Square aera).(T_T/4)


The previous diameter D as summation of (dx)correspond to a square as:

Square aera = D.D = D²

More, the space constant T_T, allows to prove a new algebraic theorem as follows.

Writting with the circle radius R we get:
(2T_T).(Somme de R_idx) = T_TR²
Finally
2.(Somme de R_idx) = R²

Theorem When a quantity (n) is associated with an elemental quantity (dx), there is a relationship between (n) and the summation process with (dx) written as:

- integral in mat (USA)

To be abble to verify this Algebraic formula we must choose good values for these data. It is done as follows:

Let L a given length. From there we can define L/n=(dx)

The more greater (n) the above formula runs well.

That is to tell
When (dx) is lesser than 0.2 then an integral and a differential equation can be of an approximation without to much error. This is valid for electronic signal processing using sampling analogic ones. However, a differential and integral calculus should not be considered as precise calculations. They are only mechanical methods to get the nearest value.


This theorem show how the Algebra is a product of human language. It can Never give an exact and precise solution.
While with construction in space, we can see immediatly the (n_i(dx)) summation as a powered variation and is a curve. Because this formula is erroneous with low (n) values where the curvature is pronounced. While this theorem is a correct one for great (n) values or very low ones for (dx). It is because at these scales, a powered funstion is similar to a straight line.

Conclusion,
It is the meaning of the summation equation (aa1).
The above summation of circumferences in a circle allows to know the circle "area". It also give the proof of the circle "area" formula.


- "Hight" + mathematics + proving + the + theorems (Russian Federation)


- 2d + space + physic (Argentina)

- definite integral formula using the method of exhaustion (USA California)

- integral calculus (Peru)

- circle + theorem (inventor) (Thailand)

Theorem of a circular disk "area". Statement

Provide that a line and a curve is a succession of elemental surfaces, the "area" of a circle is stated as follows.

An another equivalent statement of this theorem, related to the squaring a circle, is :

All these theorems need the Space Constant, only determined by squaring a circle.

The Volume of a Sphere

- Pythagorean + alphabet (Czech Republic)

The term alphabet designates the conventionnal elemental symbols representing the images or the sound that any human can reproduce to express some of his ideas during their mutual exchanges with the means of talking or reading. Then, you are telling exactely what Pythagoras tried to proceed mathematical concept as a human talking language. He did it whith with Numbers that are only conventional talking and writing languages.
Because the real intuitive mathematical notion that any alive needs to get conscience and knowledge is the pure space concept of himself and of his space envirronement. However, because the deep knowledge of space is too hard to get and also because drawing geometric figures was not so easy when lacking of drawing materials. For example, modern computer gives a great capacity to us to get more and more complex space figures. This is forbiden for the first human mathematicians. The Anciants can't but choose a way of expressing mathematics concepts with conventional languages. Thus, Numbers become the most used mathematic elements. From there Pythagoras invented the universe as Numbers. Next, Arithmetic, Algebraic and Analytic Geometry that was conscepts of space transformed into number langages, are means of doing mathematics. The basic difference between the first and the second method is the impossibility of the first to get precision in matter laws knowledge. Conventional talking and writing languages are separated with the pure elements of space by a deep lack of the first to get precision of matter laws. Precision is not an ideal of ideology. But precision is the property of space construction in the universe that is needed for human to go further into complexe constructions.


- numeral + language (Greece)

Numbers are really human invention as conventional language to tempt describing space concepts that are only length and distance existing in ratios for constructions. One can tell that space is accessed and known intuitively while numbers are human intellectual concepts. Intution is there the capacity to be conscious and to build knowledge on the universe because alive and any thing in the universe are constructions of the universe matter laws.


- object + or + natural + phenomenon + of + constant + value + used + for + reference + in + measurement (USA)


- la + sphère + pi+ nombre + énigme + forme + aussi + simple, + cachée (Belgium)

The KEY of the Beginning of the universe is the transformation of circular structure in linear possibility of object construction. That is to say construction is the beginnng of conscience and conscience is the matter of knowledge. Physically tell, beginning of construction is the possibility of measurements for interactions in construction.

From that nothing in matter space is hidden as mystery.
However when we talk about mysteries, it is only because of human limitation for taking conscience and getting knowlege on natural laws. Physical laws are simple because they are part of anybody. Thus when some physical law is difficult to know, even if it is very simple, this is due to some particular state of human thought.

On can tell that the mystery of the squaring a circle is due from a so long time where mathematicians had give up the effort of understanding the Space as matter. This gived rise the impossibilities of resolving some problems in Geometry. Geometry believes that human can get knowledge by conventional language and thus manipulates only abstract thought. However abstract thought are mostly imaginary and matter space knowledge can not be reached by theses mathematicians. Therefore for no longer, mathematics becames science of conventional language formalisms. These are the cases for Numbers, Arithmetics, Algebra and all what follow as their developments. Then, from simple natural laws, mathematics go on into the super formalism complication.

Here, the Space Constant is a KEY for a vast means for going through the space understanding. This Constant is not yet well received by the current world science and techniques becaue most of scientists may see there as a new receipts or tricks. Most part of you are interested in the old Pi of how it is irrational or how it is composed of decimals.

Here is an another importance of the Space Constant. It concerns how to get the sphere volume according to it.

Let a Cube (C) with the side equal to the cercle (C) diameters. We can operate as the obove procedure to proove the surface of a circle with the square one. Here we need to do in two times the Summatios of elemental lengths to get the cube volume. And from these results, a new form of the Space Constant give us the Sphere volume. It is done as follows :

Vol(S)<=>(Constant / 6)*Vol(C)
Vol(C)<=>(6 / Constante)*Vol(S)

In brief
According to the current notation about the old anciant's notation on Pi we can state the following theorem.

Proof on solid volumes and surfaces have more basic procedure not given here but always according to the Space Constant.

Thus the Squaring a circle is the most simple while it is the most fundamental KEY for space knowledge.


- relate + the + aera + of + a + sphere + to + the + area + of + a + circle (Duke-Energy corp, USA)

- sphere in cylinder (primus.ca, Toronto, Canada)

In Dakhiometry it can be determined probably any surface and volume of solid. This rests on rational and universal basis. It doesn't proceed as it is not done with the Archimdes' Method. This latter is no more than a mechnical one as one found usualy in empiric manner.



This is a very interesting remark.
It is an important application of form transformation parctically useful in many industrial or architectural applications.
All solid volumes are spatially proved. From there, it is very easy to do form transformations related by a same volume or a common factor.
Here is a transformation cylinder/sphere according to a common circle diameter. This is spatially done using length proportions.
An interesting corollary:
The cylinder volume external to the sphere one is equal to the volume of the sphere external to the cyclinder one.

- circular has less perimeyter than square (Pakistan)

Actually, at same volume, circular perimeter and surface aure less than the corresponding ones of a square or a cube.

- largest + lateral + surface + "area" + sphere + cylindar(USA)

Right! Largest cylinder and largest cylinder lateral surface is a cylinder inscribed in a sphere. This cylinder is characterised by a square as its section along the central axis. This square section is inscribed in the sphere. It is an important matter space fact because this space structure determines the globe characteristics of any heavenly body.
The formula is easy. This cylinder square section is the square inscribed in the greatest circonference of the sphere. The cylinder axis length and its base diameter, are the square side length.

- What + is + the + formula + for + how + to + find + the + largest + square + that + can + fit + in + a + circle (USA)

- formula + for + the + maximum + square + that + will + fit + in + a + circle (USA)

- largest + cyclinder + inscribed + in + sphere + calculus (Canada)

This cululation of the largest cylinder inscribed in a sphere is done in the determination of the heavenly globe ring structure.
The maximal cylinder fitting in a sphere is seen in 90 degrees latitude for its height.

There are many practical situations where one need to put a square form in a cylinder one. It concerns mechanical constructions as well as for foodstuff use. Then, the cyclinder circonference time the space Constant factor give directly the maximum square perimeter. From there one gets the maximum side of the square that can be adjust through this cylinder.

The proof is simple according to the following theorem of maximal product:
Let a rectangle inscribed in a given circle.
The maximum aera of a such rectangle is reached when this rectangle is a square.

For a cylinder in a sphere, let us consider its section through its principal axis of symmetry. We know that the volume (V) of a cyclinder is : V = constant.[D.h] with D=diameter and h=(axis-length).

Thus, according to the above theorem of maximal product, the maximum cylinder volume in a sphere is such that:
D = h

Therefore, it is a cylinder with a longitudinal square section.

- pie+is+round + cake+is+square + pi+is+3.14 + what+is+cake? (USA)

... Convergence towards endless insatiated goodies for wipped cream cakes ???
... Help !

Consequences :
Relationship between circle and its inscribed square.

- the + inventer + of + pie + formula (USA)

- carré + inscrit + dans + un + cercle (Belgium)

- Pi + relationship + of + circumscribed + circle + and + the + square (USA)


circle and square are linked together is positions of inscribed and circumscribed. In the case of a square inscribed in the circle, the same Space Constant is used to describe their relationships.

On this next figure are indicated the notations used in the following. As for the case of a circle inscribed in a square analyzed above, we will use the same Space Constant obtained. This method consists of using a square as reference of calculation. This reference is denoted here as the square (T). The circle has the side (d) of (T) as diameter. And the inscribed square in this circle has its side denoted as (c)
With a such reference (T) the Space constant is the same universal constant equal to (Pi/4) with its inverse value as (4/Pi).
This Constant is use to write the different reciprocal relationship between the circle and this inscribed square.

Notation : (sqr) denoted as Square Root of (2).
Then, the relationships are given in the following !

Square (T) :
P_t = 4.d ; and S_t = d²
circle :
(P) and (S)
Square (c) :
P_c ; and S_c

For perimeters we have,

P = (Pi/4). P_t = Pi . d
P_c = (4/pi) . P = 2.sqr(2) . d

Thus,
P/P_c = Pi / (2 . sqr(2))
And,
P = (Pi /(2 . sqr(2)) . P_c
P_c = ((2 . sqr(2)) /Pi) . P

For aeraes we get,

S_t = d²
And,
S = (Pi/4) . d²
S_c = (4/Pi) . S = (1/2) . d²
Thus,
S/S_c = Pi / 2
Et,
S = S_c . (Pi/2)
S_c = S . (2/Pi)

Conclusion:
It is important to generate these formula according to the method of starting from a square as reference to aobtain the Space Constant. This use allows to keep alive the method of thinking that consists of starting from a base, then unfolding successive reasons for developping construction. Remind that the circle and the square are directely linked together in succession of series. So, from a base of a square as reference, one can control the measurement of these series of figures.
In more, the use of the Space Constant from a reference, allows to easily memorize the different formula without being dependent to their formalism that will soon be a great confusion for scholastic teaching.



- impossibility + infinity + theorem

- nothing + in + the + universe + spawns + from + absolute + nothingness (USA)
You are right. The Matter Conservation from a Discrete and Finite universe, is a solid principle on that rest physical laws. Evolution is stricly matter transformations.

- pi + value + exact + convert + online + billion + value (Singapore)

Spatial proof
of the Recurrence as space property

- picture of fundamental counting principle (USA Washington dc)

In the case where the universe is a Continuum, there is no distinctive capacity for human and nothing can be counted. Counting is the property of matter discretness in absolute concrete space.
On these bases, the universe measurement rest on the logic of Succession. It is why we can know space as the Near and the Far in precise and exact manner. Knowing the Near and the Far is no other thing than knowing locations by Distance and Direction quantity. According to the discretness any Quantity is unique. It is why uniqueness of Space is necessarry know in exact and precise manner. Therefore, quantities has the property to be known Successively.
Counting is the recognition of space discretness property successively accessed by human. Also, we acknowledge motion according to the successive discretness Fact of space.

- POLYNESIAN COUNTING (Netherlands)

- fundamental counting principal (USA Virginia)

- numbers + in + displacement (Spain)
This is the just new concept for counting, quite exactly what we must understand about the human fact of counting.
While from the beginning, human was concerned with language invention for their idea communications, the Anciants until modern thinking was trapped in language systems. Thus, it is generated the known natural number series. This latter is coagulated in an artificial formalism where they tempt to given somme imprecise foundation with recurrence and induction rigid principe and then created some basis for Number concepts. It is a poor formalism figure of : 1, 2, 3,... N wher many mysteries are built around it as Incommensurability, Irrationallity, Transcendental Number also with odd things that mathematised strange knowable Infinities and Nothingness.
Numbers are ideas for movement that are facts of matter space extends. These are the only fundamental reallities that are materials for human conscience. Then Recurrence is the matter property generated by the space reality.
What does human do when he is counting ?
As for animals and for all animated beings, everyone when he is exploring his environment, need displacing in the space. It is the same basic action when he is counting. He is doing an EXHAUSTION of space. When a farmer is counting cattle, he is doing as said above. He is then exhausting this particaular space that is the total cattle space. The basic mode of counting is to use movemant to displace. To be able to exhaust, you may displace yourself or you may displace all the cattle. This the foundamental fact of how to know a space extend.
Thus the natural number series is only a very schematic language figure. And it is a harmfull habit to take this writting formalism as a mathematical model from there all the Arthmetic and Algebra are built.
The only foundemental counting or space exhaustion is to get only the real meaning of the Recurrence property of space and motion. The following text is describing this Recurrence concept.
Now we can retain the pratical Natural Number series, but we must see counting, movement or exhaustion as occuring only of space forms. There is no fixed 1, 2, 3,... But there is the basic concept of the Unit. That is to tell, There is the uniqueness of of the concept Unit but there are multiplicity of Unit forms. And it is the Reccurence that allow multiplicities. That is, the Unit is multiplicity when one tell : 1 apple, one table, one cake, one meter, one day,... ect.
Mathematical application :
1) - It is a good use for mathematics to take the Space Constant as a Unit.
2) - Also, the Unit can be the Square Root of any numeral particularly the Square Root of 2.
3) - A square or a circle or any other figure form of any dimension, can constituted an Unit.

It is why one can use an International System of units and all other different conventional measuring unit system.
En brief:
As it is shown in the following, a ONE unit can start with any dimension and then one can use it to make a space exhaustion succession. As matter is fundamentally decrete, the Recuurence is a space exhaustion succession that does never mett any hole in space. Therefore, there is no Irrational, no Infinity. The Recurence is a precise scanning of the universe fullness.



- ULTIMATE + pi + calculator (Malaysia)

- discrete + mathematic + prove + one + plus + one + equal + one (Malaysia)
You are right.
The ONE concept is the Unit. An Unit is alqays an immutable Unit. It is why at any scale there is the possibilitlity for measurement, because there is there a Unit.
Now, one apple plus one apple is equate to two apples. Because, here it concerns the quantity of a particular object.
Therefore, the Unit is the beginning of the counting or the movement process and the succession of the immutable Unit is the contuity of the matter quantity as a length or a volume,...



- situation + when + exactness + with + pi + is + necessary (USA)


- False + void + of + universe (United Arab Emirates)

- impossibility + infinity + theorem (USA)

- is + the + world + a + big + "mathamatic" + problems (USA)
Because it is absolute finite space the universe is a big Solution. While for human, because he has to walk along the knowledge Way, the World is a great Problem to solve.


- what + is + the + quantity + of + motion (USA)
Physic defines momentum. However, as recurrence is the basic principe of motion, the basic notion of the recurrence is really the "quantity og motion". Thus, the following analysis are how to figure out this quantity of motion, relative to its quantity.

- basic + parts + "og" + triangle (India)


- pi ++ pattern (Autralia)
Not only the motion aspect of the recurrence, this latter is the fundamental principe that allows construction of spatial forms. Thus, it is a just notion when telling the following recurrence relationship as "patterns". Because, understanding recurrence allows to know how there a different forms in plane and 3D figures. Therefore, the surfaces and volumes of solids can be WHOLLY deduced from this recurrence base. In Dakhiometry, all the elemental forms are precisely known from deduction, starting from one spatial base.

- construction + of + the + universe (Estonia)

- what + is + fundamental + mathematics (Croatia)

- meaning + of + space + prove (Philippines)

- pi + rational (Switzerland)

- rational + pi (Nepal)

- how + is + pi + used + in + mathematic (Hongkong)

- mathamatic + definitions (South Africa)

- foundamental + logic + equation (Hongkong)

- the + continuum + concept (Canada)
We can define the difference between Continuum and Continuity.
Continuum is and old concept that comes from the indefined space. No lacation can be measured in this latter. Thus, the impossibility to get a measure in a such Nothingness is the origin of the Continuum concept. The continuum is therefore, a monotoneous Nothing because one can't also nemed it, because a Name means necessarry a Definition. Thus Continuum means coagulated, fixed, a One as death.
The Continuity is the property of a dicrete matter space. Discrete matter means possibility of differences. This latter generates Changes. Because discretness of matter is a permanent Fact then, the Chnage in matter is an unbounded Fact. This drives us to understand that the Continuity is the property of the unbounded Change in the universe.
In brief, Continuum is the monotoneous coagualated World. While the Continuity is the property of the Change in the universe. Finally, in matter there is the basic Recurrence Property of Change.

- universe + as + a + whole (Ireland)



The unique matter in the universe is both space and motion. Matter is dicrete thus, it is necessarry a finite quantity. On what foundation does the motion come from ? It is not sufficient to tell that motion is energy. It is only a tautology. The root of motion origin is in the discretness of matter, without it there can be the universe construction. Thus we there the key of opening the door for movement. It is the RECURRENCE in matter.

Thus, the Recurrence is the expression of the fullness of matter in the universe. It the matter is unique, this means that there is nothing else than matter.
Practically tell, the recurrence means that matter can succeed in discretness, one after the other without no rupture or no fault. And when matter is in recurrence property it appears as movement phenomenon.
Nous are going there to make the proof of the matter space recurrence. The contnuity of space is in the logic concept of successive spatial figures that are independent et distintive between them. And the continuity is expressed as the perfect relationships that link together these independent figures. Thus, if a series of figures is formed successively in the continuity of a law than we have proved the space Recurrence.

Resume the preceding analysis on relationship of a square inscribed in a circle. With this elemental property a spatial series will be formed.


This figure represents a series successively formed by square inscribed in a circle. Both are linked together by the Space Constant obtained from tha the squaring a circle.

Notations:

sqr(...) means Square Root of (...).
Let denoted the Space Constant as : k = (Pi/4)

P_t = perimeter of a square (t) as reference of transformation
S_t = Aera of this square (t)

P₁ = Perimeter of the circle inscribed in the square (t), with ordinal 1 in the series
P_c1 = perimeter of the sqaure square inscribed in this circle (1)
S₁ = Aera of circle (1)
S_c1 = Aera of the square inscribed it this circle (1)

In this next notation of this series only the ordinal is changed.



- this will be pastulated (wayne.edu, USA)

- "Sqaure" + side + length + to + diagonal + length + relationship (Canada)


The preceding analysis on square inscribed in a circle, allow to establish the formula of the elemnts of this spatial series.

For the square of reference (t) :
P_t = 4d
S_t = d²

Then, the perimeters and the "area"s of each series elements can be deduced as follows :

P₁ = k . P_t
P_c1 = P_t/(sqr(2))
S₁ = k . S_t
S_c1 = S_t/2

P₂ = k² . P_t
P_c2 = P_t/(sqr(2))²
S₂ = k² . S_t
S_c2 = S_t/(2²)

P₃ = k³ . P_t
P_c3 = P_t/(sqr(2))³
S₃ = k³ . S_t
S_c3 = S_t/(2³)

... and so on.

Thus according to that the recurrence reproduces the same property, here is the Space Constant, a general formula is as follows when the successive ordinal take the one of the natural series number N :

P_n = kⁿ . P_t
P_cn = P_t/(sqr(2))ⁿ
S_n = kⁿ . S_t
S_cn = S_t/(2ⁿ)

- traducir + "precurrence" (prod-infinitum.com.mx, Mexico)

Reciprocal :
From these equations the reciprocal precurrence recurrence process can be written. Thus, from a square inscribed in a circle of ordinal (n) it can be established a same system in any ordinal order (t). The recurrence step is well ordered. It is written as follows :

P_t = P_n / (kⁿ)
P_ct = P_cn . (sqr(2)ⁿ
S_t = S_t / (kⁿ)
S_ct = S_cn . (2ⁿ)

Results:
The Recurrence process in direct et reciprocal modes in space, is fully established. In this example between a space (t) an (n) the recurrence defines the absolute Fullness of matter. The exhaustion between (t) an (n) is quite rational and precisely defined in any direction of the recurrence process. Thus we have proved there that in space the movement and its energy aspect, is completly and precisely defined, in any direction, without missing any part as fault or hole. The space is the a Fullness of rational matter.

With this series of value,
a precise Space Constant is necessary.
Thus, empirical value of Pi is irrelevent to produce this series because the circles and squares will no longer remain related between them.
Note that only the spatial figures are basic. The above numeral relationship are only used for a more familar view easyer recurrence understanding.

Remark:
In these proof are used Arithmetics and Algebra means to establish formula. But the real optimal proof can be done with the full space elements constructions.

However, this belong to the Dakhiometry method and I can't develop it there. Knowledge of the elemental operations of transformations - the addition, substraction, multiplication and division - are needed.. The reason is that the current mathematics are narrow limits that can't allow larger knowledge extends. The Dakhiometry rests on basis that one have to get without which which condition, mathematicians should remains in mathematics of receipts and tricks.

- four + sytem + to ++ "which" + earth ++ belong (Canada)


- geant ++ square + root (Belgium)
In fact the square root is also a basic space constant.
On can see that succession of squares with circles mutually inscribed, are succession of circles and at the same time these such successions of squares are successiveley done by doubling the preceeding "area". Here also, the square root of two, is necessarry rational length, otherwise these succssions can not be true. The "area" doubling will no longer be precise if square root is an irrational quantity. This space structure of square and circle inscribed and circonscribed figures, is a DIRECT PROOF of the rationality of square root.
The above recurrence proves that matter space is the fullness of matter where space can be wholy rationally measured. This fact emphasized that Numbers are really langage products that are only approximative space property representations. It is to conclude that Numbers itself can never be space materials. Therefore, incompletness of langage generates odd thing as irrationality.


- theorical + mathematic (USA)

- Do + you + favor + strict + construction + of + the + law ? (USA)
Only spatial construction can truly prove a law. In space it can appear direct abd reciprocal logic that allows a complete proof. Everything that is do not belong the a structure is seen as a false datum.

Conclusion :
The figure representing this spatial series is the unique direct fondamental proof of the Space Recurrence. And the Recurence concerns the reproducibility of the spatial relationships between circles and squares according to tha Spatial constant (k). This proof is done in two dimensions space. It is the basic law expression. It can be also extended in the three dimensions when it concerns the solids to express their volume and surfaces. This is then the proof of the total Fullness of matter in the universe. This cancel the different old hypothesis of hole, Nothingness, Infinity and Irrationnality tricks.

The different formula written here are only verification, and is given only because one use to understand the recurrence with the natural number series. However, the spatial construction is only the real valid proof.

Note that this series is succeeding in decrease order and it is denoted as the ordinal in subscripts and not on the power of perimeter and "area"s values.
In this example the beginning of the spatial series starts at any value of the square (t). But owing to the reciprocal of the Space Constant (k), the analysis can start at any beginning value and at any direction of this spatial series. In consequence, this spatial series is reciprocal. It is not necessarry to prove the preceding or the succeeding elements as it is need in arithmetic natural number series.

This spatial proof of recurrence is most foundamental for the motion knowledge and it allows an easy understanding of recurrence end motion.
Discrete matter is composed of the ultimete immutable elemental dakhion. It is from this element that the concept of ONE represent the matter unit (1) in the natural number series. It is useless to try to converge this spatial series toward some Zero that is only language convention. And seeking a zero in matter is a direct contradiction of a discrete matter principle.

There are numerous spatial series that can be proved in space. An example is given with the Numeral Generator that is given in an another page of this site. It is directely an easily constructed from space figure.



- there + is + no + void (Honduras)

A main view point you have.
The Dakhiometry Recurrence is a quite different from the current Algenraic Algebraic one. This latter is a narrow definition of the natural number series as an artificial langage creation confused with the basic matter property of motion.
The matter Recurrence is what allows movement phenomenon. Movement happens anywhere, at any location and ends where it ceases. Thus, the recurence or induction defined on the natural number series, is only a formalisme of automatic mechanic repetition that starts and ends as stated for the N series.
One can see that the Dakhiometry recurrence is here applied and illustrated by a circle-square system, at any if its origin. This is the pure basic Recurrence matter property. And the main proof that it shows is the Fullness of matter in the discrete universe. As you tell it : there is no Void in matter of the universe.

- distinguish + the + difference + between + an + "algenraic" + expression + and + &quto;&a" + algebraic + equation (USA New York East)

There is no difference because Algebraic is only a product of conventional language and is not a mathematical science of spatial structures.



- manipulate + reality + throught + quantum + mechanics (New Zeaaland)

- do + all + mathematicians + agree + on + how + N + is + defined ? (USA)
Definition of the natural number series N are yet narrow. It is clear that the recurrence and the induction principle are not very clear as concepts. This N demonstration or rather as definition are as most Algebraic proof. They are definitions or axioms that are only formalism conventions. Most of proof are not as a true proof should be.
For example, the present proof of the recurrence is a true one. This proof is complete : succession, order, contuity without rupture, directions and localisation of element that allows counting quantity. This is the Recurrence property that generates the Natural capacity of COUNTING for human when he use language to traduce the recurrence by the natural number series. It is the Recurrence that allows to understand what is human counting. It is also for tha animal one and surely in the mineral where without capacity of counting there should not the possibility of forming cristall network.

- anthropomorphism + in + life + of + pi (Austria)

- d'où + vient +l'univers (Canada)

- What + is+a + Foudamental + Counting + Principle (USA)

- set + is + infinite + if + and + only + if (USA)

... if its parts are infinite.
1, 2 and 3 are parts of a set. Any set in the universe is countable. These counts are necessary discrete to be able named. Therefore, according to this discretness, no part of the universe is infinite.



- diffrence + between + open + cycle + and + closed + cycle (Saudi Arabia)

This a quite rational differentiation of concept.
Even if matter is a finite quantity concept, the formed universe is a open cycle. We know what is the usual system control using feed-back. In these theories the open loop system is only a class of closed loop process. It is because what human can know is only closed loop system, a model of a condensed matter. It is why the Gravitational obseved on Earth is an error when it is stated as an universal law. The whole universe do not belong to a system concept class. Construction processes of the universe is an open one and is a logic inscribed in our thought.It is the Recurrence process, the life open construction.


- open + loop ++ mode ++ meanning
Because the universe existence is formed with condensed systems. Each one is a particular independant and unique system. This is according to the uniqueness of space location. Therefore, in any of condensed system, every object interior to it is linked to the other one. They are linked because of the formation mechanism that bound all the matter from an initial cloud of elemental one. Because every things are linked together with this whole system, no one can be considered as an isolated objetc. This rule is valid for any apparatus produces by man. Then, an electronic system uses to proceed by close loop method. This latter means that this apparatus is linkec to its environment where it has an input and an output.
Now, considering the whole universe, this latter is not an objetc or a system. A whole is governed with principle. The universe is the property of the Unit principle. Then, a whole is unique, it is the logic One. We can not tell that a whole is alone this expression is a language error. Thus, a whole is the logic One. This latter is only an articulation or the elemental logic. Therefore, a whole can exist without the need of linking with any environment. The unit principle allows the logic of a Whole to work without any loop, neither with his exterior nor with his interior because for the whole universe, there is no interior nor exterior. A whole is a whole. There is nothing to add in or to substract from.
With a such logic view of the unit principle, it is easy to understand now why despite the matter is permanent, the life construction in the whole universe can be a wealthy unbounded process. This is the extended concept of a simplified open loop definition.



- what + is + the + value + of + finding + rationalist + proofs + for + the + existence + "og" + GOD (Montserrat)

The absolute logic of reasonning is as follows:
If physical laws are proved and with experiences corroborated then according to logic of existence, nobody can prove the true of non-law.

Fundamental facts of the universe is its Permanence and it Beginning of construction in condensation of discrete matter. These facts are logic corroborated by observation of discretness reality. And also, these fondation are unboundless fertile for knowledge of the universe existence.

Therefore, no proof of these facts as false or untrue can be produced according to logic reasonning.

It is why the Permanence of matter and its own beginning of the construction process can't cohabit with the imaginary reality of any Creator of the universe and its whole contains.

Man, the authentic Creator.
Remind that square and circle are
immutable universal objects.

- logical + cohesion + first + cause + universe (Bulgaria)

- Beginning + of + Universe (Hongkong)

- who + was + the + creator + of + mechanical + universe? (Turkey)

- spontaneous + matter + creation (USA)

- equation + proves + no + god (UK)

The circle is before the universe and is the permanent structure of matter. Every thing and We, are made with the wealthy circular structure. Thus, each of us is not only the stars dust but more fundamentally, we are also before the universe. Therefore, the universe and each of us, are never created. circle is not god becaause it is a basic structure amongst other ones and what rests on circular structure also can't be god. Inversely, all this Whole itself can't be god because god need not to create himself when the universe and human are created. As matter is necessary permanent and this Whole is god then, a god creation is only a feed-back action in the Whole. His creation is necessary a closed loop sytem where the Output human acts on the Input god. A such god can't never be a Creator because he is at the same time created by his creation. In any way that we do reasonning on god man sticks like glue to him. Therefore, Creator and Creation are only ONE person. This ONE is not the famous UNIT but this one god is only te man hiden under a mask.

In summary, In the situation of a permanent universe where a Creator can't create a Creation when this Creation can simultaneously create his Creator then,
- A GOD CREATOR IS IMPOSSIBLE
- THERE IS ONLY MAN CREATION IMAGINARY.
A possibility of Creation is a situation of severe contradiction the existence of which forbids us to think.



- Two + only + things + touch + infinity : + human + stupidity + and + univers (Italia)


- Tao ++ how + to + draw + the + start + of +evolution (Australia)

It is an exact real fact that we can draw the start of evolution of the universe. Drawing and explaining this event is a possibility inscribed in our thought.
Exactly, at the starting of the Beginning, we all, in the past and in the futur, are being witness to this event. There was really no form because at this moment, there was not yet any form. However this event is in the permanence of the matter properties. So we can draw and understand the Beginning because, it belongs to our thought logic. It is really such. In next time, I will expose the proof of the equality of the circle and the transformed square surface. There is only simple elemantal logics. The logic of the Recurrence that allows the starting of only 1, 2 and 3 then, the proof is done ! It is simple, a supreme simplicity of the Beginning where life was born. However, this simple proof can be understood only with new concepts. What are exposed here in this text, contribute to this understanding.
You are telling the Tao, it is a just physical image of the starting universe. Because the Anciants had draw the symbole of the starting visible motion in the universe when they established the visible rotation of the universe in the form of the Yin-yang logo, due to the beginning of matter interaction for construction.


- UNIVERSE + AS + A + WHOLE (Former Czechoslovakia)

- about + the + origin + and + evoluton + of + the + universe (United Arab Emirates)

Signification:
The Total "area" (S) is equal to the "area" of an ultimate elemental matter time the total length of all the circumferences.
This is the definition of a rectangle with length equaled to the total lenght of circumferences and with the width equaled to the ultimate length of elemental matter !
Isn't it a clear definition of discrete matter "area", even if it concerns a circular structure ?


This measurement by integral calculus is only valid if Pi is known. Obviously Pi is the Spatial Constant given by the squaring a circle. Any value of Pi can be used. But Only the Unique Spatial Constant is the True.

A such process is the more exact the more (dx) value is near the utimate elemental value.


- nothingness + and + surface (USA)

This is the JUST concepts to be differentiated consciously.
We are made of and we live fully in concrete matter space. Nothingness is an old human darl ignorance. Condition of rational knowledge should concerns the real discrete and concrete matter. Many old harmful ideas and believes should be quickly thrown out. Life is not only human tricks in closed loop group business. Condition of life is to get conscience of matter environment. Human has to get knowledge of the Earth characteristics and the environment characteristic of the Earth Planet. Everything is envolving, conditions of the alive is not to be asleep in a quiet coagulated eternal fixed Earth described by dangerous belief dogma. Stop being stupid human. A NEW WORLD vision is necessary. For everyone and eachone particularly, NO ALTERNATE WAY, is the condition to survive.

Consequence : the bad process

- Euclide + straight edge (Indonesia)


Why are integrals, differentials and derivative calculus generate errors ?
Analysis are done here for understanding thes calculations modes that are widely use in Algebra to define imprecise forms with curvilinear functions versus linear variables. For example they are used for describing field forces and for standard statistic distribution models.

- how + to + find + a + "area" + of + a + circle + "id" + you + only + know + the + circumference (Canada)

... if and id only... - how + to + work + out + the + circumference (UK) - explaination + of + line + types + in + drawing + and + geometric (Belgium)


- concept + of + infinite (Korea)



- foundamental + theorem + of + integral + calculus (Greece)

- integral linear disk area (USA, New York University)

Calculus of curvilinear aeras and volumes are usually done form graphic of linear Analytical Geometry. These are only acceptable as usual approximation. However using it as Mathematics exactness is a harmful systematic error. The most use of such approximation method as for example in the Fourrier analysis, should produce many unkown artifacts and bad interpretations. It is because the Trigonometric sine function is only a specific particular mechanical motion of a round solid piece. While in the nature, there is no "wave sine trajectory". There is ONLY, parabolic wave motion. The so many Fourrier function type used in analysis is really an artificial bad systematic measurement method.
In fact, the fundamental Space Angle is really a severe science ignorance. - integration calculus formulae for multiple function (New Zealand)

- linear vs curvilinear graphs (USA)

Unhapply, it is the usual mode of Analytic Geometry.

- calculus is only an approximation (verizon, Pennsylvania, USA)

Actuallay. Because:
a) - Calculus are done only in linear algorithms.
b) - These algorithms are only forms of the famous exhaustion method.
It is useful but give only approximative results.



Resume the preceding calculation of circular disk "area".

On left of the figure, bad integral calculus process for curvilinear function versus linear variable. On right, correct integral calculus when applied only to linear function.

It is proved that minimal circumferences "area"s are equivalent to the segment ones. Thus, all circumferences that compose the "area" of the circle (C) can be drawn as segment lenghts versus the circle diameter. The next figure show it axactly as a sine or cosine function.

The next bad process is to calculate the circle "area" accorfing to a curvilinear function versus a linear variable. The first thing that Algebra does is to calculate the "area" of the circle (C) with an integral calculus applied on this function. It is the same method used for exponential equation formalism that are only cosine function types. Example can be found in the Gaus, Fermi, Poisson,... standard statistic distribution functions that are only curvilinear function versus linear variables.

These calculus methods are quite erroneous and give only rough approximations. They are empirical methods and do not allows mathematicians to verify their results. Thus, their applications are generalized in any sciences.


Now that the squaring a circle is yet found we can show how these curbilinear calculus are only eronneous from there one believes that a use of infinitesimal object allows to go anywhere in forms of space. The following calculation of the circle "area" using curvilinear function versue linear variable, gives a quite high biasing result. And practical methods that use also such algorithm gives same errors. It prove that measuring the surface "area" to determine the Pi value of 3.1416 is a fact of empirical practise.


- Squaring + a + structure (USA) - history + of ++ mathamatic (Thailand)

- archimedes + circle + triangle + "area" (USA)

- WORLD + HISTORY + OF + PI 3.14 (USA)

- summation + pi (Canada)

- positve + and + negative + about + Archimedes (Australia)

- Archimedes + "area" + exhaustion + circle (USA)

- an + equation + Archimedes + used (USA)

- enigme + horses + eggs

- WHY + IS + PERIMETER + MEASURED + IN + UNITS + AND + AREA + IN + SUQARE + UNITS (sbcglobal.net, USA)

This is a thing to be resolved. It is stange that form is associated with the Number.
From the Anciants there was known Form and Number associated in anything of the univers. Particularly for Pythagoras from what Numbers is the world archetype. Modern science comes from there. Therefore, everybody when looking to a form see there Numbers an inversely from Numbers temps to associate there physical object as sound or colors or the Good or the Evil that are also forms.
However, the universe is composed of only by discrete matter as location structures and as quantities of matter. Space structures produce forms and quantity of matter placed in these structures define its particular quantity. Therefore, a form is not characteristic of a quantity and inversely, a quantity can't tel what about its structure as form.
In conclusion
Perimeter and Aera are terms desinating both an object as quantities. A quantity is an uniqueness of any matter system that define a "matter mass" or its factor of mass. Thus, measuring a perimeter or an aera can't tell what are their placement as form in space.
But we need to introduce their forms to be abble to establish their possible relationships. Whyle a quantity is the same type of data under the unit for a perimeter however, for an aera it is defined by the square unit. It is there only formalisms of how to calculate them because a perimeter is a quantity in line and the aera is also a quantity in a line time the one in an another line. Thus, we can use a such transformation to get the final quantity of an object. For example we can define a quantity as a:
[line n time another-line)
It concerns a Quantity definition but not a Form.
It is why the xⁿ formalism induces error to mathematicians when they understand there at first as some Form manipulations. We know it in this next example:
xⁿ + yⁿ = zⁿ
Where one tempts to see there as Three or more Dimensions of Forms.

- el + inventor + de + pi + 3.1416 (Mexico)

It was the Great Archimedes' work. Here proved...


- who + invented + the + mathematic + pie + circumference (USA)

... An anciant famous unknown cordon bleu cook.

- longest + pi + value (Philippines)

how + was + the + value + of + pi + derived ?

- It + is + said + that + the + dimensions + of + the + Great + Pyramid + of + Giza + are + accurate + multiples + of + Pi, + yet + the + value + of + this + number +( 3.1415....) + was + not + discovered + until + the + time + of + the + Greeks. + How + did + the + pyramid + builders + do + it (iit.edu, Illinois Institute of Technology, USA)

It is an intersting problem to be clarifyed.
When forming large societies human capacity of building is used for building tools, memoriel and monuments. Human technology capacity began empirically. It can be known any of the elemental form et their practical relationship to be abble to project any building. Thus the circle is one of this usual form.
Without meating no theorical impossibility, the anciant architects can practically measure the circonference of a circle and clisified it as a proportional to its diameter.We may noted that a 5% to 10% of relative error is fine for building practically anything.
But it has no good reason to belieave that the Pyramid dimension is a multiple of the Pi=31.14. It is only a archeologist's hurried opinion. The first basic reason is that nobody know what is the exact Pi value prior having resolve the squaring a circle. Even if one know the true value here is a general statement about such quantity allegation.

Theorem on measurement for quantities comparison:
With (x%) of any relative measurement M,
any quantity [nM±(M.x%)] is necessary a multiple of (L.x%)
Where (n) is an integer.

Consequently
The error of Pi=3.14 is 0.1%. Thus:
(3.14)(0.1) = 0.314

Therefore
(10)(3.14) ± 0.314 = 9.9 and 10.1
And may considered as (10.Pi)
The result is that 31.4 is multiple 10 of 3.14.

This is more an illusion when M is a great number relative to the measurement error.

It is why an ant can assert that any celestial dimension is a multiple of the ant length!

Rationally tell, this problem is only the Congruence of quantities relative to a unit one.



- he + defined + the + value + of + pi + used + to + find + the + perimeter + and + "area" + of + circles (USA)

That is to say "to put the cart before the horse". Instead of trying to figure out from the circle the factor Pi as one does it when drawing milk from an udder.



- largest + value + of + pi + generated (UK)

- end + of + pi + calculate + finite (USA)

Of what Pi ? The new one ?


- discovering + the + value + of + pi-polygon-circle (South Africa)

- pi + to + 77 + billion + decimal + places (Canada)
A record but... only ???



- science + without + conscience + equal + death (USA)

- past + value + calculator (Canada) - empirical + formula + cheats

- Archimedes + struggle (Canada)

- transcendental + squaring + circle + cannot + be + done



- tortoises + systematic + errors (UK)

Nobody can't touch them. It is why tortoises are famous for their long life time...


- What + is + the + origin + of + the + expression : to + be + beside + oneself (USA)

- robot humen (UK)

- Archimedes + pi + trapping (USA)

- anciant + relationships (Israel)

- How + did + Archimedes + compute + the + "area" (Singapore)

Actually, when knowing only linear form "area"s it is an obvious temptation to equalize a curve with a straight line. The case is for the polygone and the circle. But it is more practical to divide the circle with rectangular surfaces. Such are the modern Algebraic integral calculus algorithms. More quickly, one can do it with pennies or with other elemental tiny surfaces. And really, the result are necessary biased by practical manipulations and also with systematic errors. The quite invisible systematic error is to ignore that it comes mostly from two different space structures as curvatures and straight lines. Therefore, polygons or rectangular convergent forms can never be precisely a circle. Consequently, errors is surely the destiny of integral an differential calculus. In recurrence, any Algebraic convergences and function developments know a such fate. Not only for mathematics, all the modern Physics are quite produced with integral and differential calculus. Unfortunately, it is the Continuum World fate.



- what + does + dividence + mean (USA)

- how + to + loss + weight + without + exercing (USA)

- how + to + work + out + the + "area" + of + a + circle + with + the + circumfernece (UK)

- calculate + measurement + "hight" + tan (Canada)

- How + to + invent + a + formula + that + can + guess + any + number + somebody + thinks + of (USA)

- Archimedes + determination + of + circular + "area" (Canada)

- demonstrate + experimental + methods + of + determining + the + value + of + pi (USA)


- maths +-+ pi ++ 1.34 (UK)

Really a 3.14 of 1.34 Pi. An uncertainty... plus an exact digit permutation.


The Pi value can not be obtained from polygons convergences, even if it was tell that Archimedes had do it. No sufficient precicion can be get from this process. The most probable origin of the 3.1416 old value may come from the empirical estimation of a circle surface. Then, it consists with the obtained value to divide it by the radius squared. For example, with this process, a circle surface [S] computed by digital integral calculus of 1.34% precision, one can get the following result :

Pi = [S]/(radius^2) = 3 . 14 15 927 053

- mystery + of + pi + 1.34 (Hongkong)

- Proof + of + pi + value + as + 3.14 (Papua New Guinea)

- history + of + pi + 1.34 (New Zealand)

The uncertainty value of 1.34% corresponds to a manual precision of experimental measurements. With experimental measurements one obtains always the rounded value 3.1416 of Pi. Note that the Pi value can be derived from the direct measure of the circle perimeter.


- 3.1415 + longest (Argentina)

- empirical + formula + generator (USA)

- pi + 3.1416 + et + son + explication (Canada)

- pourquoi + en + science + physique + on + met + un + seul + rang + après + la + virgule (France)



- conservation + in + measurement + lenght (Malaysia)

This is a quite juste and precise remark.
Thus the following method of measuring the circle perimeter doesn't fulfil the Conservation of length or matter space. The default here is that a circular line is refered one to one point to a straigth length (CD). It is only an empirical view and isn't a vaiable valid method.


- "vaiable" + of + relationship + in + math + field (USA)


- pi cake (Egypt)

- longest + pi + value (USA)

- longest + pi + value (Australia Victoria)

- longest + value + of + pi (University of Nevada, USA)

- How + accurate + was + archimedes + on + pi + compared + to + value + known + today (education.netspace.net.au, Australia)

- Mystic + pi! (mc.videotron.ca)

Sciences was born and is colored with Magy.

- most inaccurate value of pi (Australia)





Here is how it is used to do an integtal calculus that seems to be precise. From that one produce a dummy Pi value of 3.1415...

It will be used the altitude AB of a point on the circle relative to its diameter CD. AB is the geometric mean of the blue rectangle surface. The proof is easy to establish. Using the square diagonal CF as proportion transposition, on get directly :
CB.BD = BD.DE


The circle surface of (C) is obtained with the summation of all the AB segments when it sweeps the entire surface of (C).
We have :
AB² = BD.DE
For the purpose let (dx) a little variation of the lenght BD when it varies from zero to CD. The surface integral is then :

S = Sum from 0 to CD of
[Square root2 of (BD.DE) ].dx
The total circle surface of (C) is then S = 2S

Using the theorical "area" of circle, we get
Pi=S/[CD²/4]
It give the value of PI = 3.1415...

Doing it with a calculator, here is the algorithm:

Let (dx)= 0.0001
BD = BD + dx;
S = 2 * Sum [square root2 (BD).(CD-BD)*(dx)]
Loop until BD = CD;
The following lines are convention of programmation language type.

------------
Let : (dx)= 0.0001;
S=0;
BD=0;
Do :
BD=BD+dx;
S = S + 2 * [(BD).(CD-BD)*(dx)];
Loop from "Do" until BD = CD;
------------
Obtaining the circle surface S the Pi value is then :
Pi=S/[CD²/4]= 3.1415...

The more (dx) is tiny the more is the Pi value the one given in standard Pi used in calculator. However, with (dx) = 0.001 the precision is sufficient for usual uses.

Example, when the diameter CD=100, is divided into little parts d(x) for surface summation, one get the experimental Pi values as :
(dx)=1;   Pi= 3.1383...
(dx)=0.05; Pi= 3.141555...
(dx)=0.02; Pi= 3.141583...
(dx)=0.01; Pi= 3.141589...
(dx)=0.001; Pi= 3.141592549...
(dx)=0.0001; Pi= 3.141592651...

It is clear that dividing the diameter CD into 20 times is practically easy (case of d(x)=0.05). Even in this case, a summation of the circle surface give also a round number of :
Pi = 3.1416.


Note
With (dx) varying from (dx)=1 one get Pi=3.3836
But at higher (dx) the integral is no more valid.
One can see that empirical Pi values that was given in the whole anciant word can be sitiated there.


- Canada + calculated + complete + pi+3.14 (UK)

- Arabic geometry Archimedes theorems (Japan)

- Approximation von pi nach Descartes (t-dialin.net, Germany)

I'm sory to tell it, because in science the space means Nothingness then, Mathematics ignore the space properties as composed by linear and circular mode, the two independent space characteristics.
So, Descartes and any mathematician ignore what is "geometric" construction. The Descartes isoperimeter method is using linear convergence only for convergence towards any given defined value. It is not a rational means to get into the Circular Realm, the unknown point of circle from where a true circle radius can be defined

And more, it is like to, it seems, it hopes, but:
The Descartes' so called isoperimeter method is not a geometric construction!
A geometric construction should begin with straigth edge and compass tools. And also, it should ended with only compass and straigth edge tools. A point in a space structure should be the representative solution.
The so called Descartes' method is only a graphic that is usually shown in science. A graphic is a chart but is not a Geometric construction. Graphic are only shematic diagram as brief language replacing discours.
It is why the socalled Descartes' Pi aproximation is not a geometric resolution of problem. It is why Descartes used algebraic notation to defined the Pi approximation according to recurence mode.
Also, Descartes can't but use the Archimède pi to apply the calculation.
So,
the Descartes "isoperimeter" is only an another mathematician trick of do-it-yourself to paraphrase the Archimedes' pi.

Note
This above Dakhiometry's exercise of calculus for the approximative Pi, is not a paraphrase of Archimedes' Pi. But it is not also a geometric construction because it uses algebraic notations to define and simulate the approximative Archimedes' Pi determination.

Consequence on numeration - a + linearly + dependent + set + can + be + extended + and + it + will + stay + lineary + dependent (South Africa) - Archimedes + principle + continuum (USA) The above empirical method is a general fact in Algebra. Il is most important to know that this process is only coherent in a linear world where there is only straight lines. It is the case of the Geoemtry. But unluckily, it is the basic use of Algebra. There, one uses it to make trips toward the two Infinities and there, to drive any number out of ambush. Infiniresimals and function development series rest on such process. Thus, with : (dx) = 10-n and with (n) as any integer of the natural series, one make decimals. With (n) as greatest as wanted, infinitesimals are produced. And as (n) is unbounded... one discover irrational numbers. Note that irrational number can never be discovered because one can never find it ! It is the great properties of modern Algebra which is so happy to find a base in the Pythagoras' uncommensurable diagonal ! The error of such foundation is : One cannot belive that the goodness of an imprecise or untouched number is still indefinitly waiting that someone tries to converge towards it. It is a hopeless PRECISION definition that should not belong to just mathematics. The selfisness is to clame Algebra as a new scientific religion and thus, make mathematics coagulated. While it suffices to notice that one uses these empirical methods temporary until new advance mathematics discover more precise process of numeration. In this case, all the world is informed and mathematics researches will progress more rapidly. - Absolute + defined + space (mc.videotron.ca, Canada) Exactly. Space is a concrete absolute matter of perfectly defined location. We should make acquaintance with a such reality to be abble to build a true science done with the human capacity of deduction method. When the fundamental basis of matter are well established then deduction thinking can do any trip throughout any space structure. Therefore, it is easy to follows any universal law characteristics that are possible only on the absolute universe defined space. Trigonometric errors - what + are + the + rules + in + significant + figures + for + foundamental + operation (VietNam) Note For the Dakhiometry, the problem of angle measurement is completly defined. There is nothing to do with the quite empirical manners of Analytical and Algebraic trciks to give the illusion of the current high science exactness. There is no way to get into the exact science until one stays in the ignorance of what is the fondamental universe Space. The proof is to ask yourself, how on can construct the figure of any given angle value without using the blindness of the Number illusion. This is to tell that it is a big error to belive that the linearity of Number can't never give a sight in the space angle realm. It is clear that the squaring-a-circle has no meaning for the current official science. Because the how to have an access of any spatial angle is necessarily the way of understanding the squaring a circle. For example, from millenaries, the squaring-a-circle is for Archimedes the understanding of how to square-a-square! It is also for the Analytic Geometry inventor, the Trigonometry is only tricks of a recipe book, the way to overlap linear realm on to the unknown of the circular one. In Dakhiometry the space directions and the measurement of angle between directions, consists of how to figure out the relationship between these two fundamental characteristic of a real concrete and absolute Space. The so-called precision of trigonometry, scince the numercal capacity of calculators is only an illusion. And the decimal founded for the Archimedes' Pi is the capacity of modern science to catch at shadows. Without using such tangible knowledge of the matter space, there will be severely no exact science. Important note There is no sinusoidal curve in the nature of universal laws, as is learned in the Current Trigonometry. Even with the spatial circle construction of the Dakhiometry, it is completly proved by construction that the angular function of a circle is a real parabolic curve. Therefore, the Trigonometry is not only an approximative tool but an error of Mathematics resulting in a false angle measurement using Analytic Geometry. It is used in quite all Mathematics for precise applications particularly in Astronomy where tiny angles are very far below the Trigonometric systematic error. What is behind the Trigonometry? To be abble to have something for Mathematics to use angle, one had to chose the sides of a rectangular triangle for defining angles. A such choice is no more than empirical action because the space properties are not known in Mathematics. We can see there that only empirical basis is the foundation of Mathematics. From a such erroneous choice it is not forbiden to Mathematics to build a great realm with the Trigonometry from there modern science and technology rest. The trigonometry is not a particular exemple of modern science capacity to take a wrong way. One can see an another big virtual science with the great Relativity theories and all its derivative results produced in Astronomy. - Mia + said + that + if + you + know + the + sine + value + of + each + acute + angle + then + you + can + find + any + trigonometric + function + value + of + an + angle + of + any + measure. (rr.com, New York s, USA) That is to say Azimuthal angle measurements is known when angle between zero and 90 degrees are known, according to circle symmetry. However, how to known what is a given angle aside the multiple ones of 30 degrees. - Trigonométrie + pour + les + stupides (Canada) The generation of curvature into the straigth line and inversely, are not simply jokes. Here above is shown how Archimede is in error because he believed that polygon straigth lines can be superimposed on a curvature. Here in the following how can one figures out the distorsion when one tries to use Trigonometry for describing circular lines. It is the case of the usual wellknown y=R.sin(wt) linear function. - What does a linear function look like when it is represented symbolically (USA) - trigonometry jokes (rr.com, USA) - The unbounding function and the bounding function of the progressive construction can (link.com.eg, Egypt) - what does a quart function look like (embarqhsd.net, Texas, USA) - squareroot trigonometric function absolute value (elcamino.edu, El Camino College in Torrance, California, USA) - trigonometry illusions (Philippines, wordpress.com) - railway extends a horizontal distance, trigonometry (cantv.net, Venezuela) - trig functions real life (Pennsylvania, USA) - trigonometry difficult for university (Vietnam) - trigonometry circle + pi (Thailand, http://www.coolmath.com/) - organizational structure of RMIT Melbourne university library (Vietnam, http://www.lrc-tnu.edu.vn/bantin/sites/default/files/Tn-2.png) - TRIGONOMETRY HISTORY (optusnet.com.au, New South Walles, Australia) - trigonometry animation (Azerbaidjan) - Sydney opera house trigonometry (Australia) - Trigonometri (Thailand, =http://3.bp.blogspot.com/) - lrc linear responsibility chat (Vietnam, http://www.acm.org/ubiquity/volume_9/images/ linear_responsibility_chart.jpg) - Sydney opera house cad (Technische Universiteit, Eindhoven, Netherland) - impossible trigonometri (mack.no, Mack, Norway) - new university in Vietnam (Singapore, http://www.nikiomahe.com/wp-content/uploads/2009/05/ university-of-auckland-in-new-zealand-by-fjmt-and-archimedia-drew-4.jpg) - non rectangular trigonometry (Philippines, http://skipall.com/5z.jpg) - pastulation (Philippines) - pictures related to the history of trigonometry (Philippines, http://mrhiggins.net/algebra2/wp-content/uploads/ 2008/05/righttri2.png) - trigonometry jokes (jevc.ac.uk, UK) - trigonometry in equilibrium (The University of Nottingham, UK) The trigonometric functions can be used to approximatly describe a particular mechanical motion of a disk. It is incorrect to use them as model for general natural phenomenan for example to describe a violon string. - Abcisse + sampling + cooks (Germany, Vetter-Pharma) - Fourier (Denmark, AALBORG Universitet, Dep. of Computer Scie home at Cassiopeia) This figure illustrates the difference between an usual sinusoidal function and the TRUE relation ship between circular and starigth lines. The thick drawing line in red is the true relationship using circular into straight line transformations according to the space constant (pi/4). While in green, localised by a green grid is related to the y=R.sin(wt). Thus one can see that the true relationship is more uniform in the middle of the curve and for the Trigonometric function the curve more round passing outside of the red curve. Trigonometry starts to build ratios giving the sine and the cosine from a point on the circle. It is always a true result. However, when apply in Trigonometric determination, one can't but use a linear relation as the y=R.sin(wt). The linearity comme from the time variable. It is there the Basic Trigonometric construction error. The difference of the Trigonometric function results by an introduction of high energy frequencies. As Trigonometry is wholy used in sciences, it can produce great undefined errors. Particularly when one use signal processing with the Fourier, Laplace and other equivalent equation. Thus in Astronomy, where processing signal are highly used we may get mostly fantasy images to describe the cosmos. On the figure it appears in a fine red line the derivative of the (S) curve. The derivatived curve is splitted in its phase for a better comparison of their slopes. This forme shows that a derivative of a true circular cycle function fonction is not a Trigonometric (sin) form equal to a (cos) one. Therefore, in a Fourier sommation of signal harmonics their successive derivatives introduce necessarry both Low and High frequencies. For example, for an initial visible light radius after signal processing one will see there very much amplified Red and Violet light signals. Looking at the first derivative on this figure, it appears that in some derivative degrees the signal (S) will look like to a square signal. That is for a visible light a Fast Fourier Transform in signal processing can produce only a major Red light where the high frequency parts are there becoming invisible light. - pi + harmonic + "fonction" (Syria) Dakhiometry example of True formula for Space Directions mesurements. - trigonometric construction (Korea, EZ International Patent & Trademark Law Office) - Application of trigonometric function (India, Railnet GOV) In Diakhometry, measurements of Angles are really determined at least on its basis. Developpement of this method may be more and more in enhanced fertility by futur generations. True knowledge of space will be therefore unbounded science. - Geometry and Trigonometrie (South Africa) - curvilinear function (USA, Pennsylvania Lehigh edu) This next example of angle sine-cosine definition is really a continuous curvilinear Direction variation. The Tigonometry of angle is only a linear approximation of angle variation around a central point. Because the sine and cosine are linear segments that can't produce a true correspondence of one-to-one point with the circular line. The following example use Algebraic number notations is not the ideal function form. There is in fact a real complex space structure that is the true source of a fertile space directions knowledge. But it can't be understood easily without knowing how to spatially construct the addition, subtraction, multiplication and division operations. - rectangular + triangle + relations + between + trigonometry + formulas (Estonia) La Trigonométrie avait été montée seulement comme moyen nécessaire pour les angles. Mais les Mathématiques ne connaît que le monde référé par des axes linéaires de la géométrie analytique. C'est pourquoi la Trigonométrie porte le défaut de manquer de continuité dans sa génération de la continuité des angles autour d'un point central. Ceci, parce qu'avec une variable (x) comme abcisse sur une ligne, il est impossible de générer la continuité d'un cercle. Alors, la Trigonométrie ne peut dessiner un cercle qu'avec plein de trous. Dans ce cas, l'espace des directions, reste toujours incomplète. On ne le sait pas, simplement parce que les Marhématiques ignorent la différence entre le Linéaire et le Circulaire. La Dakhiométrie, ayant réalisée la quadrature du cercle et inversement, connaît l'espace des Directions, c'est-à-dire comment déterminer les angles avec précision, par la Règle et le Compas. Ce n'est pas facile de l'exposer parce que c'est un monde NOUVEAU inaccessible aux optiques Mathématiques, par ses ignorances et particulièrement, par orgueil et suffisance millénaires. Je vais vous donner ici un court échantillon pratique, sous forme algébrique compréhensible par vos formations mathématiques. Vous le comprendrez le mieux, sans les préjugés mathématiques sur les Nombres. Entrez dans un monde encore vierge des pollutions mentaux. Faites taire vos savantissimes formatages mathématiques alors, sans doute vous seriez prêts à être vraiment savants! - sinusoidal finesse (Turkey, Ankara Bilkent University) This next relation is a function of infinitesimal variable values (n)=dx. Starting at one (dx) to describe a 90 ° we can see that at lower angular values less than 5°, there are not continuity in the circle function. That is to say, there is no continuity when number are used even infinitesimal are used. In Trogonometry, where the linear coordinates are {y,x}, the circle perimeter continuity at lowest sinus and cosinus are nescesarrily bad. Therefore, what do we conclude for Astronomy where very low angles are measured?... - formule mathematiques du big bang (myvzw.com, USA) Vous l'avez bien vu. Pour vous dire que cette formule si simple, est à la base de la représentation des Quantités de l'univers. Il ne s'agit pas de la série usuelle des Nombres mais, de la représentation complète des Quantité de l'univers. C'est une représentation de quantité par des rapports de longueurs. Il y a une représentation des Quantités de cette manière mais, selon le mode Linéaire, qui nous est plus facile à comprendre, puisque la connaissance commune des Mathématiciens est de ne connaître et d'apprendre que selon des représentations linéaires. Mais, le merveille des merveilles, est nécessairement la représentation selon son origine qui est de le faire dans le mode circulaire. C'est-à-dire à l'aide d'une structure de cercle. C'est facile à les dire mais, le chemin pour y semer la clarté est si long et si diifficile, non pas en elle-même mais, parce que les mathématiciens ne connaissent que ce qu'ils avaient appris et ce qu'ils enseignent, toujours dans la rigidité morte du linéaire. Mais en vérité, une telle représentation de toutes les Quantités de l'univers par un cercle est un autre veritable Big Bang. Car, il faut gravir notre robotisme et automatisme, une montagne certainement plus haute que celle de Mars, une montagne comme celle de la Terre en Carré, un relief à côté duquel la Séries Infinie des objets imaginaires de Dedekind, n'est qu'une platitude d'un terrain de tennis tout neuf. Voici une idée de comment générer les angles, si l'on ne sait qu'utiliser le vieil langage des Nombres: Cette formule génère, à la fois, le sinus et le cosinus du complément, des angles de 90° à 0°, rien que cela, par exemple, de mesures d'angle en degré. - Avec la vaiable (n) qui est un chiffre variant de Zéro au Sans-limite, en quantités continues de la matière discrète. Il suffit de renormaliser les variables pour obtenir les mêmes égalités classiques de la Trigonométrie. Faites en la vérification, cette formule donnera la continuité de tous les angles d'un quadrant de cercle, allant de 90-0° pour n variant de toutes les valeurs décimales possibles. Evidemment, la racine Zéro-ième est une erreur de calcul. Simplement cela est dû aux méthodes algébriques de calcul des racines, par tâtonnement en exhaustion de divisions. Mais cette formule simple, permet de calculer à la fois, tous les arc-sinus et arc-cosinus. Comme simplicité de la Dakhiométrie, il n'y a pas mieux!!! Elle donne directement le sinus et cosinus de tout angle dans le quart de cercle directionnel de l'espace. Avec l'Arc-cos on obtient la valeur de l'angle selon l'unité choisie. Cette fonction génère la continuité des angles de l'espace. Cependant qu'avec les moyens Algébriques, il y aura toujours des tous de valeurs, dus à la non-continuité des Nombres Algébriques. Les calculs utilisés pour vérifier cette formules se font avec les modes appliqués de Trigonométrie, inclus dans les calculateurs. Les angles générés par cette formule souffriront des défauts des méthodes de la Trigonométrie. Dans sa vérité, la définitions par mesures des angles, ne sera qu'une merveille que par l'espression des constructions par l'espace. On peut alors avoir une vue d'ensemble d'une structure complexe qui donne une vue complète des angles. C'est de cette structure que cette formule algébrique ci-dessus est tirée. La Dakhiométrie est à même de montrer la haute valeur du véritable Penser humain. Ainsi, lorsque je disais que les Mathématiques et la Physiques des sciences en cours ne raccontent que des bobards simplistes de primitifs, ce n'est pas parce ce que "j'aime ou j'aime pas ceci ou celà". Les sciences en cours ne sont que des fantaisies de "savants". Ce n'est que des paroles du Penser rationnel que chaque homme est encore incapable de deviner sa propre capacité... mais malheureusement bien momifiée et plein de suffisance. - cosine + law + prove + concrete (Canada, Alberta Edmonton) - examples of real life square root function (USA, Florida) - computer resolution circle drawing (Korea) - big crunch diagram (Columbia) - GALAXY creative Vietnam (Singapore) - weird science animated (USA, New Jersey) - equation describing a circle (UK, NHS Health) This equation can describe a circle continuously. Continuity here depend to the division algorithm of modern calculator But is more fine than the usual Trigonometric sine and cosine in an orthogonal co-ordinates reference axis. - prove + the + existence + and + uniqueness + of + a + perpendicular + drawn + from + a + point + P + to + a + line + L (USA, New York, Poughkeepsie Marist College) Uniqueness of a perpendicular to a line L, is directly from the structure of a rectangular triangle inscribed in a circle. The two rectangular sides of this triangle are unique lines defined by a circle. In addition, the above formula of angle sine-cosine show that any angle is uniquely define by a circle. Therefore, a perpendicular from a point to a line, is necessarry unique according to that any angle non rectangular in a rectangular triangle are unique. The usual angle functions determined by the Dakhiometry - use of trigonometry in daily life (India) - transformation from circle to angular (Singapore) Right! However, it is only because angular quantities are directly the circle property. Contrary to the Trigonometry that introduce errors in the angle definition when it is done according to the linear Analytic Geometry axis. - different symbol use in trigo (Thailand) It is here shown under particular formalism. But when expressed un full spatial language, this latter is an fertile enlightening on the precise angle characteristics. - thales trigonometry (Australia) These are only formalism resemblances. One can see it only when they are expressed in pure spatial language. If a such trigonometry can be built with the Thales rules, all of the world mathematicians of the anciant times probably had done yet a such system. There is really a more basic reason that allowss to get a so fertile view point. - Importance + of + having + the + basic + knowledge + in + trogonometry + to + the + college + students (pldt.net, Philippines) This following formulas are the basic ones for angle determination. They are very fertile and easy in applications. When calculation with spatial elements is applied with them then, the precise angle properties can be unfolded by deduction in extended wealthy space structure. - why + do + all + the + angles + in + a + trinagle + add + up + to + 180 (eiu.edu, Eastern Illinois University, USA) It is a basic remark. Now that the angles are determined by a point (A) on a segment (BD) then the answer will be more precise. A triangle is only a brief of a circle circumference. The total triangle angles is only situated between 0-180 degrees because the circle angles are completly determined by the point (A) in (BD) can situated only between (BA=0) to (BA=BD). It can varies back between (BA=BD to (BA=0). This is a form of angle variation stricly between (90.2=180) degrees Because the linear is born from the circular then, the basic reason is from a fixed point, two intersecting lines can be never more than being perpendicular and make an angle no more greater than (90.2=180) degrees. We should note that the perpendicular line is then the property of circle angle. A Triangle can't have no other angle property than the circle one. - nature + of + mathematics + "pastulates" (airtelbroadband.in, India) At least a great pastulate... The Dakhiometry had proved and spatialy constructed all the functions to determined completly the basic angles between 0 and 90°. These proofs rest on the basic properties of the absolute space the ones unknown in Mathematics. The angle determinations are as follows: Let any determined length (BD). Define three colinear lengths related with (BD) as: BA + AD = BD Circular knowledge consequences. Fulness of space-direction determination method in Dakhiometry - the properties of angle in circle (Malaysia) Actually, only and strictly, circle properties can resolve the space Direction and then, consequently, the linear realm. You are informed that the Dakhiometry method of dtermining the space Direction - usually denoted in the classical Trigonometry as Angles -. They are done more simply, perfectly precise and with exactness. Any angles are defined by the above ratio (BA + AD = BD). there is no need of sine and cosine definition and no more their inverse as Arc-sin for example, to expressed them in degrees or in another unit. For example, An angle is given as the ratio conventionally named as follows:          ... etc And it inverse derivative Arc-cos is directly given by simple spatial construction. This give the direction relatively to a reference one. So let a given angle (a) in any choosen unit, you can figure out precisely a such angle. Also, from a given angle (a) in degrees for example, it can be simply constructed its form as this above "cos" ratio. That is to tel, with any ratio (BA/BD), one can figure out an angle without calculating the Arcsin or Arccos that are only approxiations. And given any angle, one can determine the ratio (BA/BD) corresponding to this angle. Any angle value are rationally spatial constructed. and the usual addition-substarction-multiplication-division can be used to transform this give angle. For example, one can construct any inscribed polygones in circle with any given vertex. Note on errors in Trigonometry: The current Trigonomety can do better angle definitions than with a precision of 10%. It is because of the Archimedes' Pi systematic error that is no more precise than to the first decimal. How can Astronomy and Cosmology can be built on such hugue errors? Here follows are exemples according to the trogonometrc conventions for a best easyer understanding. Then, we can define from there the angular continuity of functions for angle (a) situated between 0-90° as follows: Examples: Let (a+b)/2=BA3 and (a-b)/2=BA4 then, is written as: - FRENCH + CLASSE + SIN + MORATUWA (slt.lk, Sri Lanka) - rational + values + of + the + cosinus (rima-tde.net, Spain) - CIRCULAR AND TRIGONOMETRIC FUNCTION (Philippines) - SEEDING SLOPE FORMULA (twbeaumont.com, Time Warner Cable Beaumont, USA) - applications of trigonometry story problems (ncren.net, USA) It is usually known as Trigonometry however, here is represented a brief introduction to spave Direction determination known as angular measurement. It is quite complete precise definition of angles with its basic tools for angle transformation. Because space Direction is from circular realm quite different from the Linear mode. This latter is only originated from the initial circular mode therefore, the linear can be only some practical approximations with irrationality imprecisions due to Numbers as strictly dichotomic language words. INFORMATION: A complet and precise Angle determination in Dakhiometry You are informed that Space direction or Angle is still unkown in Mathematics even if it was created there some apporximation with the empirical Trigonometry. The Dakhiometry had achieved a complete precise and rational scientific system of defining angle, according to his full knowledge on the space circular realm. Angle is rationally determined according to precise space constructions. It shows that the Trigonometry has a certainty no better than 1e-6. It is not included there the classical Archimdedes' Pi with its uncertainty at 1/10 of unit ( 3.1??? ). This proof of errors is determined by spacial constructions which is characteristics of absolute precisions. It is not easy to simply expose it here because a such system need deeper basic space knowledge that mathematicians has no access. It consists to know how to calculate space structures with the space language then, to know how to manipulate the circle and any conics figures. We have to consider that Mathematics Geometry and Analytic Geometry is really simplistic when compare to the Dakhiometry science possibilities. Important difference: Trigonometry uses Cosine and Sine for defining angles according to a linear orthogonal axis reference. While the Dakhiometry Angle definition use direct circular definition of circle arc. There is no sine and cosine and their derivative conventions. This allows to use the four transformation operations on circle arcs or angles, as addition, substraction, multiplication and division. Circular lines are then completly equivalent to linear lines for any space structure construction. For a simple example: it can be done by using space construction, the Angle Trisection and more, to divide an angle by any quantity. Any circular polygon with any integer number sides can be constructed as a circumference divided by a natural Number. Note that this process can't never produce a circle as was tempted by Archimedes. Note Such possibilities are one of the most important consequences deduced from the Dakhiometry squaring a circle solution. Example of errors on some angle measurements On figure of the squaring a circle it can be get the angle that allows to determine the form transformation. Let this angle denoted as AS. In the following are measurement differences between trigonometry and Dakhiometry. Mesure with the classical Trigonometry method as it is used with a CAD sofware this angle is given as: 1) ------------- Trigonometry: ; Dakhiometry: ; With the Pi value 3.1416... of computer: ; 2) ------------ Trigonometry: ; Dakhiometry: ; With the Pi value 3.1416... of computer: ; 3) ------------ Trigonometry: ; Dakhiometry: ; With the Pi value 3.1416... of computer: ; - trig use in cad (westottawa.net (West Ottawa Public Schools, Canada) There are current illusions in modern technology where anything are determined by digital machines. For calculations, one attributes precision with math-coprocessors long word digits. But it is an illusion when the main action is controled by the softwares. These latter are no more than the usual Mathematics algorithms that rest on the approximate circular functions. For exemple the Achimedes' Pi value in conjunction with the current Trigonometry introduce error until 1/10 in any calculation result. Warning Any of CAD sofware used in the whole word contains the Archimedean Pi (as 3.1415...). As circle are highly used in practical CAD designs, paid attention that even if math processors used with long floatting digit precision there are many great errors during calculations for figure constructions. It is because most of figure in modern technology concerns circular forms. The introduce errors ares not visible because every calculation rests on this same incorrect constant. For example if you measure your house with a same false unit of 1.5 meter length, you will not see there any error. However, you need an external another unit of 1 meter length to be abble to see that all your anterior measures are really erroneous ones. The Pi=31.1415... is a bad value for high precision mechanisms. All the Dakhiometry calculation are done by space construction and not with the Algebraic operations. It is needed only to divide two lengths then to multiply them by an final length. No classical Trigonometric formula is needed as for example the arc-sine function. It is noted that measuring the space structure is done also directly with the Space Constant length as: Space Constant = 3.120 193 73 It is seen that the old Archimedes Pi introduces error until the unit angle. Measurement Conclusion: Measurements on figure lines are more precise with a minimum use of digital number calculations. Therefore the result with the Dakhiometry method is better than the uncertainty of the Trogonometry plus the numerous step for calculation using particularly the Euclidean division. Therefore, the errors from Trogonometry is seen there with no better than 1/1000 decimal. This can be obviously worse ones when theses errors are repeeted during a long algorithm repetition of initial errors. Therefore, Trigonometric angles are not as exactness warranted from second of angle precision or les than 1% degree precision. Science errors are the most serious modern problem that international human society has to rapidly resolve. Nothing can't goes further in optimal evolution ways when it rests on false actions. - application of trigonometry (airtelbroadband.in, India) - trigonometric proofs (tbdsl.com, TampaBay DSL, USA) Note What are shown here are only first sight. The Dakhiometry circular angle definitions are complete. Angle can be manipulate for precise transformations as it can be done in linear mode. Important Sailors use to search their displacement direction according to stars with the sextant navigation instrument. This latter rests on a three axis coordinates principle. This is a method irrelevent in three dimensions extraterrestrial space because there is no terrestrial horizontal as given plane reference. For the Dakhiometry method of spatial angle determination, it needs no more the sextant principle. It suffices to determine the direction target relatively to any fixe given reference direction. For example this latter is determined by a faraway star that can give a sufficient good direction of reference. It is considered there that the parallax of a faraway star is equivalent to a fixed direction. Then the linear straight way toward the target is known. All the trigonometric formulas can be easily verifyed with these above equalities. For example, the (cos 2a) can be extended in its general form as (cos n.a) It is impossible in Trigonometry to express a such relationship because of its formalism complication. But in Dakhiometry, a simple spatial structure can define the multiplication of an angle (a) construction. Its allows to construct a polygone in a circle without worying with the successive reporting of a same arc or chord. Note also that in Dakhiométrie all the four basic operations can be done on the sine and cosine, for example: (cos a)/(cos b) is easily done. It suffice to read the result from the space structure. The continuity of the sine and cosine should be done with these relationships and not with the angular velocity as (w.t). Spatial Direction in three dimensions - cosine curve in three dimensions (USA) Angle is the space characteristic of Direction. From there any angle in three dimensions is determined by this above formula even if it is given here only a shematic formula of the Dakhiometry for angle determination. Therefore, any angle is determined from one given space direction as a fixed referent angle direction. It is the BD direction as referent. A space direction is a straight line direction. It is the case of a spatial vessel standing at a given absolute position point (B). From there it can go toward any other given direction according to the above angle formula from relationship (BA+AD=BD). Now obviously it can be defined any direction 3D space, by choosing two or many other different determined directions as referents ones. To go a distance at any precise point (D) it must be known precisely the distance (BD). Basic tool to define an agnle - trigonometry basics descent (USA) Yes. Here, space direction rests in steady deep basic sources of space properties. It is more precisely, clearly and completly defined in spatial language. Angles are there defined not with linear Numbers but by spatial elements of discrete space. That is, angular directions are defined as space points seen from a given fixed point. Angle can be only defined by circular manner. However, we human can only have access to linear capbility. As in Analytic Geometry, angle is erroneously defines by two coordinates of orhtogonal lines, named as sine and cosine. The Dakhiometry can spatially prove that the circular angle can be represented in using relationships between three colinear line segments as: BA + AD = BD In this case Any angle (a), for example between 0-90°, can be define by a length (BM). This latter is used to figure out the particular angle (a) by the spatial language forms. It is written as the following: Angle (a) = BM = sqr(BA.BD) The curve as a result, is seen in this next figure, comparing to the usual Trigonometric function: y = x . sin(wt); or with cos(wt) In Dakhiometry, the "angle=(BM)" is not a function of a so called angle variation but is a quadratic function of the segment (BA). The curve is denoted by (1) relatively to the (2) one of the usual Trigonometric function. We can see on this figure that there is a real difference of the quadratic function with the linear Trigonometric formalism. The (BM) is on the Y axis while the variable (BA) is on the X axis. - sound in parabole (Cyprus) Sound is necessarily due to movement in parabola form. Also it is for any vibration in nature. The reason is the universal law of absolute space where potential energy governs the space location of any object. A potential is composed of a defined rest point location. Kinetic energy is then the consequence of potential location from where material object cyclic changes around its rest point. This is the universal characteristic of absolute location in space. Then sound is material atomic and molecular matter that move around their local rest point and also of system of these elements that is orbiting around their rest point according to the potential versus kinetic energy law. One should consider that heavenly bodies orbiting not in perfect circular orbits are also vibration whith ultra low frequency. For the earth for example, the cycle is one year duration. But we can consider that these duration is no more the vibration scale and be considered as movement one. Heavenly bodies remains in conservation energy because they are independnent object by matter condensation process constructions. Thus, the Newton Universal Gravitation is a contradiction where a Big Bang and a Big Crunch principles are contradictory to the Energy Conservation in the Whole universe. An absolute space has no expanding nor crunching mechanism and the Relativity in space is only anthropomorphic empirism of historic bad tale. - sine wawe to square wave converter (Latvia) A such electronic instrument is consistent with the sound phenomena. A loud-speaker is really functionning according to such function. Probably the speaker working will less suffer the resonant and distortion effects than with the classic mode of the Trigonometric sine function produced by current synthesizers. - Algebra and Trigonometry jokes (saratogaschools.org, Saratoga Springs City School District, New York, USA) Really they do. Because there is a precise approach of definition when using the properties of space. The element of space language can do it with an easy understanding of problems when space strutures allows to defined their curvilinear an the linear aspects. - creative sine function (brookdalecc.edu, Brookdale, The County College of Monmouth, Lincroft, New Jersey, USA) - trigonometry on curved surfaces (Korea) - Sydney Opera house hyperbolic paraboloid (Ireland) - what does a convergent function look like (teksavvy.com, Canada) - trigonometry comparison (Indonesia, http://psycnet.apa.org/) - parabolic trigonometry (pacbell.net, California, USA) - fungsi grafik trigonometri cosinus (Indonesia) - how + does + Trigonometric + form + of + a + complex + number + apply + to +daily +life (tmodns.net, USA) - trigonometry in medical image (USA) Modern surgery using computer 3-dimensions images aid needs not only exactness but precise point determination e.g. is the brain surgical operation. Precise 3D directions is there needed. - vision the library university RMIT (Vietnam, http://ultibase.rmit.edu.au/Articles/dec97/crock3.gif) - grafik trigonometri sin 2x (Inonesia, http://www.mathwarehouse.com/trigonometry/ period-sine-cosine/images/) - closed form expression acos(wt) (strath.ac.uk, University of Strathclyde, Glashow, Scotland, UK) - Egypt quadratic function (Malaysia, http://wpcontent.answers.com/wikipedia/ commons/thumb/d/d1/Quadratic_equation_coefficients.png/) This figure show that all this above function are quadratic ones. The sine or cosine curve are denoted there by (1) and the red curve denoted by (2) is the Trigonometric wellknown [{sin(wt)=y]. We can see that there are errors from this latter compared with the Dakhiometry ones that are parabolic functions. Note that this above formulas are only simple transcriptions of the Dakhiometry's spatial structure towards simplistic Algebra numerical formalisms. But in their spatial structures it can be read directly the sine or cosine of an angle and also they figure out directly the given angle. That is to tell, in a spatial structure an angle is not directly read only as its sine but also as its inverse denoted by the Arc-sine. - equivalence point from second derivative curve (ntu.edu.sg, Singapore) - describing sine and cosine (umn.edu, University of Minnesota, USA) - animasi trigonometri matematika (Indonesia) - graph sine 0-90 degrees (iplannetworks.net) Example of the coagulated Mathematics What can Mathematics know about a parabola? this latter is there diescribed as defined by a director axe and a focus. From there complicated algebraic functions tempt to defined a such parabole. Here in the following we will see what is a parabola according to the Dakhiometry. - ordonnées à l'origine parabole (Canada) - parabolas in real life (tds.net, Telephone & Data Systems) - parabolas in everyday life (tenet.edu, Texas Education Network, USA) - MOTHER PARABOLA PROJECT (rr.com, Texas, USA) - Mathematical parabolas (Germany) - parabola equation math college.org (rdsnet.ro, Romania) - general parabolic equation (ic.ac.uk, Imperial College, London, UK) - mathes formule Iran - accurator parabola (Indonesai) - Iran maths (Brunei Darussalam) - parabol arc quadratic equation (internode.on.net, Vietnam/Australia) - mathematical definition of parabola (ocn.ne.jp, Okinawa, Japan, http://www.onemathematicalcat.com/Math/Algebra_II_obj/) - maths tool Singapore (Singapore, http://3.bp.blogspot.com/ _iszZx0ntq_Q/SokofUWCezI/) - parabola feed (Korea, http://www.radio-electronics.com/info/antennas/ parabolic/parabolic_reflector.gif) - parabola ends of lateral (Philippines) - Foyer parabole (ocn.ne.jp, Tokyo, Japan, http://www.ilemaths.net/img/forum_img/ 0258/forum_258808_2.png) - parabolas in real life (Cocos (Keeling) Islands, Costa Rica) An unique particular parabola and more generally, a given parabolic trajectory, can be defines precisely with unbounded space parameters. Thus This figure shows that a given parabola (P) can be determined by a focus (F) as the center of the series of cocentric circles and any isoscele triangle. One of this isocele triangle side pass through (F). Its other side should be parallel to one unique given direction. The parabola point is then defined by its vertex resulting from the intersection of thes two sides. This figure describes how these isoscele triangles are positionned when the fixe direction of one of the isoscele triangle is parallel to it symmetric axis. Therfore, a) - A parabola is not generally defined by a focus and an axe director. b) - A parabola should be defined by concentric circles as Focus and by a fixed direction. - for + every + angle + there + is + axactly + one + angle + bisector + true + or + false (nycmny.east.verizon.net, USA) It is true. Absolutely, we have to take any angle as the unit one. Then we have to anlyze this problem as follows: In the Natural Interger series, what can be divided by TWO? Therfore, from an unit angle, only EVEN quantity can have a bisector. It is only a true concept over the universe Fact. This the rational logic applyed to the dicrete matter: only even quantity of elemental matter can be exactly divided by two. However, in our condensed matter world, we can't never have the elemental matter exactness. It is why in the particular exactness of the condensed matter approach, we can tell that ANY ANGLE has NECESSARRILY AN UNIQUE BISSECTOR. Dakhiometry as a true new science Matter Space has two fundamental characteristics: 1) - Angular or Directional type 2) - Linear type 3) - The two characteristics are unified according to a Space Constant for mutual Transformation. The old Mathematical Geometry are not space science but is only empirical approach to some incomplete superficial formalism. While the universe fundamental basic first reality denoted as space is fully mastered by the Dakhiometry science. Space is quite known and practised through its linear and curvilinear aspect. Practising space is a clear conscience on Measurement. It is based on: a) - A just conscience on Quantity and how to determine it. b) - Measurements are concerned on the two space aspects. It is how to determine: Linear and Curvilinear Quantities. Knowledge on Space is a quite fullness of Dakhiometry conscience. That means: the Dakhiometry practise a true basic Science Language that is composed of basic space elements. The current official Mathematics have no means to get a little sight in the Dakhiometry science because the current official science all along the Time and through all different epochs, ignores what is Space. Accuracy of angle according to the Dakhiometry method Property of space point is to be successive. It concerns for the Linear realm and also for the Curvilinear one. In Dakhiometry a precise mode of Angle determination is to correspond point-to-point from Linear to Curvilinear. This means that the ratio determined in the Straight line can be applied to the Curved line. Therefore, as we can do on linear realm all the four calculation operations, we can applied the same possibilities of calculations in Angular realm to masrer constructions of any spacial figures. Thus, linear calculations operate on distances and curvilinear ones operate on angles in space. The precision is the one of lines intersections. Note * On a straight line, points determine distances. * While on a circle, points is directions -or angle- determination. Here is an example of Mathematics ignorance. Everybody believes that he know what is an ANGLE. However: *   Mathematics ingnore the three basic modes of space determinations: the squaring a circle, the trisection of an angle and the doubling a cube. This is a proof of a quite space Ignorance. *   Science ignorance is shown on how it ignore the Quantity and what is the Counting. Numbers are only a Holly Trick for chatting that give to men the illusion of telling science. *   Trigonometry is a same trick to overcome the Angle ignorance. According to that Mathematics don't know how to square a circle then Mathematics ignore what is true or false about Linear and Curvilinear data. *   Consequently, was born the Analytic Geometry. It gives the illusion that science is mastering the measuremnt of the World and wield power the Nature. A simple example on analitycal geometry: with a so called y=ax2, Mathematics prove that a parabola is an Ignorance. This seems an immoderate expression, doesn't it? So, do you know that a parabola is a pure space form as a circle, a square, a triangle,...etc, are a pure space forms? Here is what is a diagram of successive conic space structure: - real life parabolas trajectory (cox.net, Oklahoma, USA) In consequences, do know how to place anywhere in a given space structure, a parabola or only one part of it? Mathematics have no means to do it else than with illusive manipulations. Because knowing what is a space conic else than its analytic forms, you can tell that you know what is space! Ignorance of Mathematics gives rise to ignorance of all the different Physics Théories where everything are weird harmful tales for human. In conclusion In addition to the Linear World, the Dakhiometry masterises also the Science of Angle in the true universe Space.

The mathematics infinitesimal belief

Discussions :

Conditions of space transformations is necessary a reversible process.
In this case, these transformation should be done according to the following properties.

A line (x) represented by infinitsimal unit (dx) seems to be confused with any curvature (C). If this statement is true there should be a property continuation linking line (x) to the curve (C). There is thus a relationship represented by the way (1).
In this case, a such relationship may be valid in the reverse way (2).

However, aside some particular cases of function,
there are many results of such transformations given by the uses of integral and its reverse derivative calculus. These results are :

1) - An integral of a linear function versus a linear variable, gives ALSO a linear one. There is no continuation along the way (1).
2) - An integration of a curve give ALSO a curve. There is no continuation along the way (2).

3) - A derivative of a curve is ALSO a curve. There is no continuation along the way (2).
4) - A drivative of a straight line is a constant. There is no continuation along the way (1).

Note
One may remember that the squaring a circle is a reversible transformation. Therefore, in space, curve and straight line can be linked.

Finally here is a remark about the Infinitesimal object property when it is used as integral and derivative calculus. Generaly, for these purposes a curve function is used with a linear variables e.g. as (dx) or d(t). Then,

- The integral and derivative functions cannot give exect exact results according to that a linear variable cannot describe exactly a curvilinear function.
- The property of a curve and the one of a straight line, do not depend on the scale that one consider them.
- Curve and straight line are different spatial structures.
- More precisely tell, a linear VARIATION variable does not correspond, one to one, with the curve ones. Even if there is a one-to-one point, it can't be possible to get the complete correspondance between linearity and curvature.


- "exect" + measurement + for + pie + 3.14 (Brunei Darussalam)

These conditions concern all the infinitesimal calculus, include the developments of functions. Using these methodes are rough approximations when applied on curves. Ideas of using tools as infinitesimal means to reach any spatial structures are quite false. The above example of circle surface calculation proves it. Fundamentally, it is quite an error to believe that infinitesimal objects may allow to reach into any spatial structure.

The important usefullness of these results, is to show that really there is a belief concerning the INFINIRESIMAL mathematics methods. Believing that any infinitesimal displacement in a straight line is equivalent to a displacement on a curve is the false axiom in mathematics. Remind that infinitesimal is the basic tools for Algebra.

Finally, there is really a key that allow entering the circular form. the Squaring a circle is the ONE and also the UNIQUE ONE to open the circular structure for spatial knowledge. It is not exaggerated to tell that the Sqaring a circle is the key opening to the universe space knowledge.

- unit + circle + for + "calulus" (Canada)

Comments:

It is possible to obtain the circle "area" value in doing measurements and also with the integral calculus. Note that all the integral calculus calulus and its derivatives are empirical methods.
This two empirical ways don't need to know the Pi value. The squaring a circle was regognized as impossible then, it is straight forward to invent the Pi value as the circle "area" divided by its radius squared. One can verify this invention in doing a such measurement or using integral calculus. From that, it is find the Pi value exactly the standard one used in the whole world.

The most interesting result is that this 3.1416 EMPIRICAL VALUE, is raised to a Trancendental an Irrationnal number !

Empirical means are useful. However, one can't paste on empirical method a Transcendental and Irrational concepts. It is a contradiction of incertainty concepts that transforms thing into miracle.

Now, this Pi value obtained by empirical method, is the result of many square roots and Euclidean division calculations, according to Algebraic methods and principles.

Therefore,
On can conclude that the above error value of 1.34% on the surface calculations or measurements, are a general characteristic of the imprecise Algebra method.

- He + defined + the + value + of + pi + 3.14 + used + to + find + the + perimeter + and + "area" + of + circles (USA)

Quite exact ! He put the cart before the horse !!!

- archimedean + axiom + wrong (Croatia)

- pi + used + in + inertia (USA)

Actually, Pi should be an inertia symbol of wasting one's time on never touched goal as the Achilles' Tortoise.


- what + number + did + the + pythagoras + fear + since + it + lies + between + the + only + two + integers + that + can + be + both + the + perimeter + and + "area" + of + the + same + rectangle (Canada)

It was anciant and is presently metaphysical human psychology. The most important problem is not to find what a number is or is not. It is to take conscience of difference between spatial structure of the universe and human language. In anciant time and also today, space did not exist but there is only nothingness. Relativity fields are not space. They are objects replacing the ancient ether. Knowledge of space was masked by the natural capacity of human to produce his language. Thus, lookning in nature, it is natural to observe that things can be counted. But between things, there is always something. A such "something" is inexpressible by words of langages. As nature is filled by gods, it is obvious to attribute this inexpressible things by irrational concept in language. However language is not Mathematics. Therefore, in the futur, mathematicians have to invent how to build a language capable to "discribe" describe the space with High Fidelity. At this time human and also mathematicians wil have no more reason to be fear by these mysterious "converge but never touched" beings.

Is Pi an irrational number ?

- "WHER" + THE + MATHEMATICAL + INVENTIONS (Vietnam)

... if negatve result with Robotsearch then, somwhere is nowher...

In the absolute space no irrationnality value is allowed. This is due to the foundation principle of discrete matter laws.

Now, because it is presented here a Pi formula where on can see there full of the square root of two symbol, then thinking automatism of mathematicians bow to the irrationality of the new boen number of Pi.
No, Pi is not transcendent nor irrational number. So, because a so much persons automatically to the look of formalism, I show here an another formalism for Pi formula and I am waiting for reaction to this new formula formalism.

- What + is + use + to + describe + the + motion + of + a + body + "aroud" + its + axis (USA)

- impossibility + squaring + the + circle + proof + of (student.uu.se, Välkommen till Uppsala Universitet, Uppsala, Sweden)

- full + pi + length (USA)

- why + is + impossible + for + squaring + a + circle ?(USA)

- squaring + the + circle (Belarus)

- mental gymnastic circle rectangle (Portugal)

Actually. A not so obvious one... - which one is height and which one is length in rectangle (Canada)

A real anxious problem. Why a North is the North and a South is the South, a 1 is 5V or an On and a 0 is a oV or an Off?

- a circle and its tangent which show the properties at the point of tangency. (Carolina, USA)

- squaring + the + circle + to + high + accuracy (swbell.net, USA)


- transformation + of + a + circle + into + a + square (ua.edu, University of Alabama, Tuscaloosa, Alabama, USA)

- in a square diagram what is length and breath% (United Arab Emirates)



Here from (A) to C1, it means a length of space structure attached to square root of two.

- "http://eberry.fr/NEVER SHOUT NEVER" (bnf.fr, Bibliothèque National de France, France)


Resume to this figure:
Let S_AD the greater square surface of side [AD] et S_AB the one of the little square of side [AB]. The ratio of these surfaces can be written as :
S_AB/S_AD = Pi/4


Calculation of Pi lenght is done around the initial circle of diameter AD. Then le lenght Pi is as follow.

If the little square is the transform of the circle [C]. Then the Pi value is directly given as follows:

Pi = 4..(AB²/AD²)

or :

Pi = 4.(x²/y²)

Also, we can write :

Pi = 4/(AD²/AB²)

You can use these two formula for Pi calculaation concerning any dimension of figures. ANY DIMENSION is valid to obtain the Pi value given in the following formula full of square roots of 2. This means that any number in the whole universe is "irrationnal". That is really a miracle !!!

In brief:
These equation are quite rational. And they prove that the following formalisme is also RATIONAL. And consequently, Square Root of Two is necessary rational :



This last formula was determined to prove also that the numeral Pi is a universal constant INDEPENDENTLY of the surface formes. It is only a surplus of proof because considering the surface ratio as ratio formed by squared numerals then, the constancy of Pi is enough explicit.

- boo + boo + boo + square + root + of + 2

Therefore, out of any ideology, who can find some reason of the irrationallity in such a formula ? Because the numeral Pi cannot be at the same time, both itself and its contrary.

- iF + TWO + POLYGONS + ARE + SIMILAR + THEN + THE + RATIO + OF + THEIR + CORRESPONDING + SIDES + AND + THE + "RATION" + OF + THEIR + PERIMERTERS + ARE (konnect.net, Konnect Internet Service, USA)

- "ration" + between + diameter + and + circumference + of + a + circle (USA)

In the opposite case, everything is irrationnal, even the integer natural number series. So, according to this formula, the Pi value can not be an irrational one. Because [4] is a rational number and the ration ratio can be written with any possible value. In consequence, if Pi is irrationnal, all existing numbers are also irrationnal.
The above formula proves that
the statement of "Pi is irrational" is an unescapable contradictory one.
Conclusion :
Therefore, Pi is necessary a rational numeral.

Now, there are many who complain or enjoy that the numeral Pi seems not finish to calculate. But don't enjoy with it, I am sory to remind them that they don't know how to divide. And also, they ignore that they fon't know it. To enjoy more note that most of numerals division can't never finish to an exact value.

Remind that the division use the Euclide's division algorithm. This process is generally not an exact dividion. Mathematics don't know another method to divide. The reason is of great funfamental one and even for a mathematician of genius, Pythagoras is a magnificient example, we must know that : numeral is not mathematic but only obey to human language rules !


- what's + the + square + root + of + one + third + pi ?

- how + can + you + proove + that + the + square + root + of + 2 + is + irrational (Canada)

- What + is + the + reason + to + find + the + end + of + pi

This is the main question.

Predict + pi + nth + decimal + place ?

- "FORMILA" + DE + PI (Mexico)

- "area" + of + a + circle + "formila"

- numeral + handwriting ++- kid +- kids

- how + is + pi + determined + so + exactly

Does squaring the circle done when the perimeter or the surfaces are equal between them ?

Inportant
What must be said clearly about the squaring a circle.

The squaring a circle can not be done in the equality of the perimeter between the circle and the square !

The squaring a circle can not be done according to their perimeter equality.

Exactly,
The squaring a circle is done under the following unique condition:

Both, the perimeter AND the surface of the circle

are equal

to only the square surface.

Inversely,
Transformation of a square to a circle can be done initially with the Surface OR the perimeter of the square, to find the Surface of the transformed circle. Reason of these conditions rests on the unique correspondence between a numeral and its square and inversely. Example : let [x] the length of the square Side OR the circle Diameter. Then :

4.x = x²
and thus :
4 = x

This result is unique and immutable. And the squaring a circle with its reciprocal, are done always with this factor [4] that is a part of the constant denoted Pi.

It is why in the Dakhiometry, it is only considered the figure of a circle Quarter inscribed in a Quarter of Square.

To remember it :
TWO values of the one corresponding to ONLY ONR SURFACE of the other.


- "perimetr" (Gorgia)

- structure + of + matter + "expalined" (Canada)

- Why + is + there + also + + Pi + in + the + "perimetr" + formula ?

This is a main point to understand why there is Pi. From that Pi appears as a space constant and not as a particular effect of a circular form.

Pi is only a proportional factor that appears in a homothetic space transformation. This is explained as herefafter :

A figure "area" is the result of two lenghts product
k.(x.y)
and can be write as
kx.y                    (00f)

Then, any form of plane figure can has their "area" written as under the above relationship (00f).
For the square :
with k=1; kx = y;    it give: y.y = y²
For a rectangular surface :
ky.(y)
For circle surface:
ky.(y)

And a circle known form is :
ky.(y) = Pi.y.(y) = Pi.y²
in that k is denotted as follow : k.y = Pi.y
Therefore Pi appears really as a length homothetic factor.

For their perimeters
Square perimeter = 4.(y)
Rectangular perimeter = k.4.(y)
Circular perimeter = k.4.(y)
and in a more usual form :
2.Pi.(y)
k.4 is denotted as follow : k.4 = Pi.2
Pi is also a homothetic length factor particularly applied to circle because circle and square are invariable closely related forms.

In a product of lengths to describe different forms, it can be use a same length for all this forms and the remainder can be seen as a different homothetic lengths. According to that for tha square, rectangular and circular forms, it can be seen that they have a same length (x). Their differences are then "discribed" described by other homothetic ones as described above.

Pi is only a homothetic constant used for circular figure, generating from a square form.

This is due to its Geometric origin. Because Pythagoras had begun to know the square. He applied it as the basis of his Geometric developments and also for his number theory. But the fact that any form are derived from the square is only a very particular case known by Pythagoras. From this epoch the World is formed with some given particular forms that are only hypothesis built from a limited knowledge on the universe.

In conclusion :
Pi is usually known as attached to circular figure but Pi is not a property of circles. Now, we have to understand Pi as only a constant factor use for homothetic space transformation.

- Indian + mathematician + created + a + formula + to + rapidly + calculate + the + value + of + pi (from .us)

How + to + figure + Pi

Fig-4-4
From a circle {C} inscribed in a square of side length AD, one can obtain a litle circle {C1} charactérizing this spatial system. Placing {C1} at the center {O}, a segment [AB} tangent to it can determine the side {AB} of a square. This latter is the squaring of the circle {C} and has a surface equal to the circle one. The Lam-Ca theorem shows that the surface of the rectangle colored in cyan is also equate to the cercle {C} surface. In consequence, in this rectangle if the side {AD=4} then the other side {AB' = Pi}.

Results :
1) - The cyan rectangular surface is equal to {4 . Pi}.
2) - Examining the formula {4 . Pi}, one can see that it represents also both the surface and the perimeter P of the circle {C}, when its radius is R = 2. According that :

P = {2.R.Pi} = {2.2.Pi} = {Pi.2²}

Inversely

From a length of 4.Pi, one can obtain any rectangle having this surface value. Using the Lam-Ca theorem one gets a square representing this rectangular surface. Then, the reciprocal of the squaring a circle can give the circle {C}. This circle {C} has necessary the perimeter {4.Pi}.

Note that any value {q.Pi} is also valid. Therefore, any fraction of Pi is a measure of a straight segment and a perimeter of circle, provide that any number is a square surface and any other derived forms as a rectangle of the figure-4-4. We can state it as follows :

Theorem of the equivalence
between a circle perimeter and a right segment :

Any circle perimeter can be precisely straightened and inversely, any straight segment is an exact circular perimeter.

Fig-4-5
After construction of the length equate to Pi, as done above in figure-4-4, we can see in this figure-4-5, the construction of any perimeter of the universe circles. Each perimeter is equal to the surface of a corresponding rectangle.

On this figure, any perimeter of circle C_i is equal to the corresponding {x_i.Pi}, that are the product of the rectangle sides, {x_i} and {Pi}. This construction is valid starting from the quantum matter to the maximal value of the matter in the universe. All these rectangles haves a constant side equal to Pi.

The figure-4-5 shows the rectangular surface, with the length Pi as a side, measures the circle perimeter. Inversely, a circle perimeter, is the measure of a rectangular surface. There is an exact one-to-one correspondence between these two figures.

Conclusion on the circle perimeter :
1) - The constant length Pi is rational.
2) - Any rectangular form can be constructed from a circle perimeter.
3) - This agrees to the Dakhiometry Third Postulate on matter that states any object number can be considered as a square surface.

Therefore, from the starting of Evolution, it is why a thread can be coiled and uncoiled from a spool and a wheel can run on a road. One may say that it is obvious. But I think that any woman and man mathematicians necessarily have a secret question about this "length concordance" since the time where a circle perimeter was famous to be incommensurable with a straight line...

In summary:
The Lam-Ca theorem alows to define in the circle-square spatial structure a quantity that can be in four forms : the surface of the circle {C}, the surface of a square of side length {AB}, the surface of the cyan rectangle and also the Perimeter of the circle {C}.

Any perimeter length can be rationally contructed!

- algebra abstracts figures (Puerto Rico)

- geometry continuity (USA Wyoming u)

- rectangle and circle architecture (Australia)

- width Theorems and Postulates (USA)

- fantastic + facts + about + pi maths (United Arab Emirates)

- circle perimeter (USA Florida edu)

- Dessine moi un cercle dont la circonférence est un nombre rationnel (Belgium)

- "squere" root of (telkom-ipnet.co.za, South Africa)

- square roots sculptur (club-internet, France)

En effet, au compas et à la règle, comme les maçons et le tailleurs de pierre.


The problem is how to construct any circle perimeter equal to any given number particularly equal to any Natural Number.

It is esposed here a simple Dakhiometry method proving that any circle perimeter can be constructed. That is to say, any numeral can be spatially constructed. In consequence, there is no Irrational Number in the universe.


Here is the method, step by step reasonning, of how to construct any cicrcle circle perimeter.

Even if this construction is very simple, it prove that from millenarries the mathematical world is emprisoned in his dummy irrational ideology.


We will see here how a circle of perimeter equal to 3 can be rapidly constructed. The numeral 3 is know as a rational Natural Integer Number. It is an example that any number or more precisely tell, any space length is constructible.

1) - We begin to construct a square surface equal to 3.

From the basic universal theorem Lam-Ca we can do it as shown in this next figure.

Only the Lam-Ca theorem allow to rationally do this construction.

2) - Next, we need to construct the Space Constant in squaring a circle C, with the little circle C1. On this next figure, this construction is shown in blue lines. We get the Space Constant length usually denoted as Pi or Pie.

3) - With this referent Length as diameter we draw a circle Co.


- Representation of irrational number on a number line (India)

Actually.
Any segment on a straight line can be a diameter for a circle. Then, any Number measuring this diameter can be represente a constructible number. More any square root of this diamater can be also constructed. This means that any irrational number can be constructed on this line "of Numbers". Therefore, there is no irrational Number, provide that all of them are constructible.
This is for particular constructions with the circle property. But there is in Dakhiometry, a specific spatial structure where constructions of CONTIGUOUS defined points proving thet there is no irrational hole between adjacent and successive segment lengths. This valid for any numeral base even for binary and hexadecimal ones. This shows that every numeric base are only conventional things for man-Numering-counting. The true Quantities are the one constructible an definable by spatial elements.


- wawe hot (Vietnam, http://farm3.static.flickr.com/2713/ 4431175665_470fb4f732_b.jpg)

- construct square root of on the number line (utep.edu, The University of Texas at El Paso, USA)

- Only the Lam-Ca theorem allow to rationally - calculate by Iteration sqrt(3) (Israel)

- The Square Root of (163data.com.cn, Hong Kong)

- geometrically construct root (uni.edu (University of Northern Iowa, Cedar Falls, Iowa, USA)

- "simplefying + squair" + root + example (Canada)

- NATIONAL SQUARE ROOT DAY (nbsdefaultservices.com, NBS National Bankruptcy Services, USA)

- get real pi .. be rational (University of North Texas, Denton, USA)

- square construction methods gif (Algeria)


- Circle arc integration (Germany, Schaeffler Group LUK)

- does a circle equal a square (purdue.edu, USA)

- perfectly equalled sides (broadband.hu, Hungary)

In the circle Co it can be placed the previous square of side equal to square root of 3.
Now with on the basis of the LamCa theorem, we can transform this square surface onto a rectangle of same surface, using the square of the side equal to the Space constant length.

Solution
We get finally the circle of perimeter equal to 3. It is the one the diameter of which is equal to one side of this rectangle.

Any circle can be rationnally constructed.
Therefore, the Space Constant is rational.


Remark Reasons of these construction rest deeply on the universe principles more than the ones named here. That is to say, this construction and its results are quite rational.

Consequence
Any quantity of circle perimeter can be spatially constructed. More basically tell, there in no Irrational Number.

This present construction is only a one that show how there is no Irrational Number in the universe

Important
The Dakhiometry can do such construction from any given quantity beginning from Zero to unlimited one. You may verify that Mathematicians can do a such complete resolution because there is no fundamental bases in Mathematics.

- inverse square root calc (USA)

- number + series + exist + in + value + of + pi (Pakistan)

Fig-4-5-1
Here are the basic characteristic of the spatial structure of a square. This can be shown by the two circles C1 and C2. The surface of C2 is two times the C1 one.
The squaring of the circle [C] is thus precisely determined by these two characteristic circles [C1] and [C2]. Therefore the squaring of circle [C] is given by the side [AB] of the resulting square. And the precise length [AB']is the length of Pi value.

Actually, AB is tangent to two circles [C1] and [C2]. The reason is that the point {A], is the center of a hmothecy of factot h = (Square Root of 2) of the latter two circles.

The point [B] allows thus a precise and unequivocal intersection point in the square, for the squaring the circle [C].

Also, the angle [bAb'] = 27,96875154...° is the characteristic angle of the squaring a circle system. Il is talk about it in the following texte.

According to these square characteristics, the structure of a square allows the fundamental precise and accurate knowledge of the matter space. The knowledge property, or the physical interaction of matter, did not happen in the central symmetry motion. In this universe state, the EXISTENCE of the Perpendicular Fact was not set up.


- how + to + figure + circumference
- The length of PI
- delimited pi
- does a circle come from a polygon

- Religious + Science + belief + death (USA)

- anthropomorphism + in + Life + of + Pi

- Why + has + the + world + recognized + a + value + for + pi + for + longer + than + the + existence + of + irrational + numbers

How to figure out the square root of (2) and the square root of (Pi) lengths, with the compass and stratght edge

The length
SQUARE ROOT of TWO is the
FIRST UNIVERSAL CONSTANT
of rational construction in the matter space



- "costructions" to the future (Italy)

- square + root + of + pi

- the + power + of + theorem + of + circle

- application + of + new + pi + formula

- square root of two (Canada, British Columbia, Burnaby, Inst. of Technology)

Note that square root of 2 shown here is not due to the diagonal of a square. It belong to a whole space structure of any ith-root quite precisely determined as construction. Because if a line is supposed to represent the current Reals Numbers series, in space construction, this same but real line can be described as a contineous nth-root in a continuation of discrete quantities. The example is shown above to define any angle cosisne by nth-root numerals.


- shows + that + square + root + of + 3 + minus + square + root + of + 5 + is + irrational + using + proof + by + contradiction (pldt.net, Philippines)

- different kinds of spatial figures with pictures and meaning (Philippines)

- project k12 calculation of pi (Turkey)





Here is the spatial calculation
of [sqr(5)-sqr(3)]. According to that irrationnal number definition is not constructible. This calculation is here quite constructible. I can't find any contradiction. There is irrationallity when algebraic method is used.
Therefore, there is a contradiction when one proves that this substraction of square roots is irrational.


- how do you answer A square with length 5 km (tpgi.com.au, Australia)

On this next figure, if the unit will be (1 km) then, length of 5 km is the diameter of circle (C5). Any other unit can be deduced from there according to homothetic forms.
Homothetic property is that it changes Distances in proportional forms but don't changes space Directions.
Homothetie is the direct property of the Central Symmetry postulate.

- circle theorems maths (Brunei Darussalam)

- square root (ntlworld.ie, Ireland)

- square root day (Biatorbágyi Kábeltévé Kft, Hungary)

- National Square Root Day (USA)


- geometric construction square root of two (USA, The University of IOWA)

- naturally occuring square roots (ca.us, California)

- ROOT LENGTH DENSITY (Korea)

- square roots in ancient Indian mathematics (Australia)

- Cosmos square root of two (purdue.edu (Purdue University, West Lafeyette, Indiana, USA)

In fact, any quantity from zero to the universe greatest one, can be writtenunder square-root form as successive discrete continuity. This shows that Numbers are strictly conventional formalism of words language. Numbers characteristics is that they don't obey and be expressed in true successive continuity.


write + a + program + to + approximate + the + value + of + PI + using + the + formila + given + including + terms + up + through + 1 , 99 (comcast.net, California, USA)




Hereafter are figured out lengths of Square root of 2 and square root of Pi.
These lengths are rational provide that they are results of compas and straight edge constructions. These constructions are here given according to the circle Theorem, given latter in this text.

Fig-4-5-2

The figure 4-4 is resume here to show how the square root of 2 and the square root of Pi are obtained. These legnth values belong to the number Q series.

The diameter of [C} is equal to 4 and the one of [C'} is its half that is equal to 2. You may mesure and verify that AE and AF are these values :

AE = Square Root of (2)
AF = Square Root of (Pi)

This is because :
AO = 2 x Square Root of (2)
That is : AB = 2 x Square Root of (Pi)


- hard math pie (comcast.net, Minnesota, USA)

- really hard math equation with sqaure roots (mchsi.com, USA)

- Giant confusing math equation (east.verizon.net, USA)

- longest math equation (USA)

As exposed here, this equation seems to be long. It is only because this method is a new one and have to be decomposed to be easily understood. But it is the most shortest way of calculation. Because how do we explain this following algebraic calculation formalism if we have to exposed it to an extraterrestrial being provide that he never knows what are our methods and tricks for calculation?

For example how do we explain this next formalism to kids biginners in math?
This next spatial calculation is done rapidly in the most short way if we are currently in use with the new spatial method of four basic operations as addition, substraction, multiplication and division. It is because the spatial language is an universal reality. It is not built according to so many particular human languages done with artificial conventions.
Thus, it is expected that in the futur, kids can rapidly learn, mastering and inventing spatial solution of most complicated technical construction problems. Until now it is not really the natural phenomena that are complicated. What is complicated is only the human empirical sciences that rest on their languages coagualted in formalism of algebraic numerical calculations due to their ignorance on fundamental facts of space.
Knowledge of space reality is in itself the action of humnan living.

- impossible math equation (USA)

In memory of Anciant math we can actually denoted it as the Impossible Equation.

- equation hard maths (Chile)

The difficulty of this present equation is not due to the spatial construction. It is because we are coagulated by the Ancient's math dogma. This latter is an empirical mode for decomposing things "world by world" as man use to "analyse" everything. Thus, old sciences are only analytical ways of understanding phenomena. You can note that any anciant proof of math theorem are done according to mathematicians' tricks of the trade. Rationally tell, these theorems are only verifications of remarkable figures.
A rational proof should be done from upsteam of any problem as for any river that comes necessarily from a source. For rational science, any construction should come from the perfect definition of the space. A construction is only possible if it rests on a space structure.
Therefore, understanding a structure means that our thinking should overcome the old empirical mode of problem analysis. Empirical analytic method can not but growing in complication of tricks and formalisms. Understanding a space structure should be a global or intuitive way of thinking.
Difficulties of resolving problems are due to sciences formatted by empirical mode of thinking. With such habit space structure seems to be Impossible thing. However, space structure is the real basic rational tool for human thinking capacity. Space properties are the most simple and wealthy natural language for human knowledge... only when old science dogma disappear from our habits.

- constructible numbers (ucc.ie, Coláiste na hOllscolle Corcaigh, Univversity College Cork, Ireland)

It is the rational number definition.
However Numbers are only language words to name the fact concept of quantity. Thus, numbers are reducing manner due to the discontinuity of language symbols to be abble to represent the continuity of the natural quantities. When Numbers are seen as the world physic archetype its discontinuity obviously seems to describe a world with full of irrationality holes in the universe.
In fact there is no hole in the universe physic. While there is really the science Number system full of holes according to millenary systematic errors. The Numbers are really inaccurate method for science. Obviously, human need language words but science should to reconsider that factors of reality is the quantity.
In Dakhiometry, it is seen that the universe space is characterized by the Distance and the Direction. Then, what is in space are necessarily characterized as a system formed in a quantity of locations. From there it is spatialy proved that the concept of quantity can be perfectly defined with the space properties. Thus, any quantity in the universe can be constructed in the continuity principle of the discrete space matter. This continuity of quantity is prooved with spatial element construction. Spatial construction resolution consists to resolve an unique spatial point as the intersection of two linear-curvilinear lines as a precise quantity of space locations. Therefore, there is basic rationally proof both for linear and curvilinear modes that there is no irrational hole in the discrete universe fulness.

- root size of Cleveland pear (quickclic.net, Canada)

How to get precisely your pear.

- actual + accurate + value + of + pie + .edu (Singapore)

- rational + new + pi + accurate + value (Netherlands)

- formule pi (Glaslokaal, Den Hagg, Netherlands)

- pythagorean ideology (sky.com, UK)

True accuracy of value OP shown on this next figure, is only the property of spatial language. But if one uses it as numeric digital then there are errors introduced by the bad Euclide's division algorithm and sources of error due to the Number Theory. All these systematic error rest on the Pythagorean ideology of Numbers.

- a picture of what Pierre de Fermat invented/the french number theroy (optonline.net, USA)

- pi + accurateness (epfl.ch, Lausane, Switzerland)

- square root: Descartes' calculation (Serbia)

- EXPLANATION how was pi derived (Philippines)

Spatial calculations are straight from a given structure.
Thus, according to the Absolute space properties, a given structure defines by itself the relationships between points and lines. These relationships concern the quantities of points and segment of lines.
Therefore, what is for these quantities, a spatial structure explains by itself how to get a given quantity of a segment for example, that is related to its structure environment.
It is why the calcultations should be use the spatial method to deternime how from a given structure some quantity is related to it placement in space relatively to its particular structure where it is inscribed.
The squaring a circle, is then the only way to get the correct structure environment to determine the exact relationship between a circle and its another form non curvilinear as a square. This is the unique way to get the relationships between the Curvilinear and the Linear, these two basic aspects of the Absolute Space.
The Dakhiometry is composed with a complete method for spatial calculation in any complex manner needed. These calculations can resolve simply and precisely what is impossible for the usual Algebraic technique. The Numbers is only man's empirical means that give to the current science a millenary false idea about what is the quantities of the universe matter.
This next calculation for the Space-Constant (the old false empirical Pi) is a simple example.





Length of Pi is calculated latter in this text as R4= (square root of Pi).
Then R4 intersecting with the tangent OM by OT is the square root of Pi.
This next figure shows also how to verify that
OT² = 2.(OT') = OP = Pi

- how + can + a + rational + pi + change + the + world + of + computation (Canada)

To avoid error of algebraic formulas calculations, it can be choose to determine the figure lengths that can determine the new Pi value with only the usual relationship:
k² = x² + y²
where there are no need of the Euclidean division approximations.


- who + invented + the + accurate + measurement + of + pi (South Africa)

- what + present + impact + does + the + internet web + have + on + mathematics (fios.verizon.net, USA)

Surely.
For example considering my personal case, I was during thirty years a laboratory assistant where I was only a simple dumb machine. Aside this situation, never any of my own work exposed in this web site was accepted by any scientific journal for publication. If there was no internet possibilities, the present squaring a circle remains still unknown and may be, for ever it remains as the glorious possibility of Mathematic Impossibility!
But still from year 2000, sciences are not yet intersted to replace the present precise Space Constant and was still prefering the old empirical erroneous Archimedes' Pi. In brief, this current world is really living a nigthmare and is not so optimistic.

- square root math jokes

- math equation solved with a miracle (kc.rr.com, USA)

Actually, if overcoming difficulties in rational research on how to determine and calculate with spatial elements is a miracle... :-)
The results shown here are only the consequences from the knowing of how to do the four basic operations of spatial calculation. Any complicated algebraic formula can be done with such method but with rational precision and exactness in space structures. To get numerical result it suffice to use only the welknown rectangle relationship formula without suffering approximations of the Euclidean division for the number roots.
Squaring a circle is not a joke due to Hazard. It rests on a vast background of deep knowledge in space properties. The Impossibility Theorem for the squaring a circle is the example of how Mathematics ignore what is space. It proves that Geometry was stillborn mathematician vows and became rapidly formalism of human dogma.

- giant math equations (bellsouth.net, Miami, USA)

Note that the four operations for calculation using spatial elements need at first to chooe any conventional unit of length that allows then to fefine length quantities according to this figure unit.
This is not an artificial convention but rest on the basic logic of successive quantitities of the ultimate elemental matter. In fact what is known as Induction principle for theory numbers is done on the base of the unit ONE (the 1 number). But mathematicians ignore that this ONE (the 1) is only a necessary logic representing the ultimate element of quantity. From there the Integer Number series can't never rest on the base of a ZERO. Zero is not a Quantity and therefore, should not be a Number.
According to this Number theory starting necessarily from the (1) the Arithmetics and the Algebraic need not to initialize a defined unit to go on calculation. Because Arithmetics and Algebraic are only in one particular case of Conventional Number beginning from the Number 1.
Spatial calculation of space structure needs at first to define a unit of length. It is because space has two main characteristics:
1) - Directions
2) - Distances.
In Direction one can access to the whole universe direction space. But in Distance the whole universe space should be define by homothetic considerations. This last space characteristic gives rise to the need of chosing measurement unit because homothetic realm is composed of different length scales.


- volume sqrt quarter substract (it-austria.net, Austria)

- MENTION THE PROPERTIES OF SQUARE IN MATHEMATIES (Niger)




- MEASURE PI CIRCLE (Korea)

- C1 symbol (Philippines)

- grade 11 circle geometry (telus.net, Canada)

- struture + curve + of + in + Cambodia (Cambosia)

- concrete circles geometry (Puerto Rico)



- creative circles, geometry (Bahrain)

- geometry of solar cooker on coordinate axis (Ethiopia)

- geometry + theorum + and + pastulate (east.verizon.net, New Jersey, USA)

- big tangent circle (ppppol.de, Germany)

Circle (C1) is really a basic little Big circle in the square-circle structure. Here is how to inscribe a square (Q) in the circle (C) using the (C1) as referent with point (P and P'). Also, how it determines a 8-gones.


Proof that the tangent AN to circle C1 intersect at point B to C2 and C

- Usa north carolina The Art Institute of Charlotte (orange.md, Moldavia)

- hyperbolic tangent circle (The University of North Carolina at Charlotte, USA)

- proof of a tangent meeting the radius at 90 degrees (virginmedia.com, USA)

Here is this Proof:
Let any circle (C1) of center (O) and radius (ON) and a tangent (AN). Any tangent to this circle is defined as intersecting (C1) at one point (N).
Now let (N1) and (N2) the next preceding and the next susccessive points of (N) on the (C1) circonference.
Because the preceding and the successive point of the discrete mater, are equal in distance then:
(N1;N) = (N;N2)
They are two unit arcs of circle (C1). This means that the two formed angles are equal:
(N1;Ô;N) = (N;Ô;N2)
This is a reason to state that a tangent (AN) to a circle (C1), has the radius (AN) a symmetric axis.

Now, from Zero to 90 degrees, what is any existing angle formed by the radius (ON) and the line (AN) that can have the radius (ON) as symmetric axis, provide that they have only 1 common point at (N)?
Result:
1) - Zero is not an existing angle.
2) - While exclusively, two 90 degrees angles can symetrically disposed around the axis (AN) because a line (AN) is one direction with an angle of 180 degrees.

Conclusion
Tangent Theorem:
Any tangent to a circle intersect the coresponding unique circle radius always at 90 degrees


- TANGENT TIL CIRKEL (stofanet.dk, Stofanet.Mine Sider, Denmark)

- crown radius and rolling radius (ucdavis.edu, Davis University of California, Davis, USA)

ON and OB characrerized by a square of side OB. - big circle and small circle with same tangent (Indonesia)

- when you trisect a triangle internally the intersection provides the vertices of a smaller traingle. what ever the original shape the smaller traingle will be equilateral. it shows the relationship of the random to the definite. (airtelbroadband.in,India)

Really, there is no possible relationship between random and defined property. If a random is well defined, it is the negation of random. Inversely, defined can't never produce random result. They are two physiqcal properties taht can't never replaced each other. Physic are governed by the Uniqueness principle. That is to tell, for example, we never observe an object falling upward without the aid of a rational mechanism.

For the case of the trisector, it is quite governed by defined construction. If it seem to a process that produces defined from random it is due only to the fact that the Morley theorem was dicovered in inverse manner. The little defined equilateral triangle inside a greater triangle, seems to be the consequence of trisector constructions. It is only the result of the usual mathematics empirism method that proceeds only from consequence of a law to state this law and not from basic reason. In fact the Morley theorem should be start from the true source of trisector construction that should be done from any triangle property. Its final result should be the trisector construction and not the inverse that is the little equilateral triangle.
So, this Morley triangle should be really starting from the defined to reveal the defined equilateral triangle. There is no random. This latter is only the virtual world to designate what we ignore in the universe. For example, Mathematics has invented the Irrational Numbers because Mathematique ignore what is the Quantity uniqueness. Only language worlds is creating as Numbers to simulate the missing Quantity concept. This is the fact of irrational belief replacing the rational basic quantity.
When one know how to construct a triangle by constructing all its trisectors then one can construct any n-trisector of a triangle. Note that these constructions are additive operation and not dividing one. From there the internal little equilateral triangle appears only as a particular case of the trisector construction.
In conclusion
There is more much high interesting thing to learn than to repeat any of the empirical Mathematics trick product... Because without the space language, Mathematics can't do nothing else than to turn around a vicious closed loop.

Here is the proof that tangent AN to circle C1 occupies a specific position in the structure. It intersects at point B to the two circles C1 and C. Tangent AN is constructed with the circle C3.

It is shown above that aera of circle C2 is two times the C2 one. This the base of this proof. We will consider here the square of these two circle radius.

Therefore, in the triangle OBN is necessarry a rectangular one because the square of side OB is two times the square of side ON. And because the square of side BN is a right angle, the triangle ABD is also a rectangular one.

This is the proof that tangent AN intersects at point M the two circles C2 and C.





- biblical signs two circles (TimeWarner-telecom, USA)

- given two circles construct a mutual tangent to both (USA)

- history of unit (Vietnam)

The how to define the unit in space construction allows the mastering of spatial calculation, the perfectly precise and exact method for rational space manipulation. The Da-khi-Hoc is actually mastering a such method.

- David Brown base tangent comparator method (India)

The solution is given when we see that all circles are homothetic forms. Thus, circles C1 and C2 are necessarrily homothetic forms.

This next figure gives the solution according to successive constructions numbered 1, 2 and 3. Beginning with the C1 and C2 homothetic center (M) determination with their two parallel radius (r1) and (r2). Then, the tangent (tg) of both circles is found with a rectangular determination with the magenta colored circle.



- the + length + of + a + rectangle + is + 5 + meters + more + than + twice + the + width + Express + the + lenght + "og" + the + rectangle + in + terms + of + the + width (USA)

Length versus width of this rectangle is given by the blue line on this next diagram.

Irrationality of a unit square diagonal is the expression of an incommensurate universe ignorance of Pythagoras and modern Mathematics

- unicerse (Mexico)

The unknown universe...


The circle is the reminiscence of the matter permanence from that the unicerse is endless constructed.

The above little circle C2 shows that the square root of 2 is inherent to the circle. That is to say, the square root of 2 is a Permanent universe quantity. Because matter is discrete, the square root of 2 is necessarily a precise quantity.

It is shown here the reason of how a square root of two is precisely determined, along a line formed by a successive localisation of the ultimate elemental matter.

- formula for diagonal of square (USA)

- Euclide to Pythagoras (India)

- Pythagoras secret society writing above door (tds.net, TDS)

- Secret Society 60th grade (tele2.se, Sweden)

- physics measurement of length using set square (HongKong)

- common tangents of two circles (Indonesia, http://jwilson.coe.uga.edu/)

The square structure allows to construct (C1) and (C2) in space to get square root of 2 length. Therefore, this latter is a rationnal and universal quantity. While a diagonal of a unit square is a dummy object because one cannot know what is a length=1. This latter is not a unique value.
A unit length must be defined as only a convention.

- art-square root of 2 (Point Loma Nazarene University, San Diego, California, USA)

- root lattice (Australia Sydney u edu)

Necessarry the discrete space basic latice structure.

- square root of 2 on a number line (Hongkong)


Resume the squaring a circle construction This figure is how to construct the circles (C1) and (C2). circle (C2) has its aera two time the (C1) one. They are includes in the circle-square structure. As all circles are homothetic figures, (C1) and (C2) are homothetic circles with the center located at point (o) of the square corner.
According their aeras the ratio of lengths is :

OO₁/OO₂ = sqr(2)

Therefore, along a universe line we can get any ratio of lengths equal to the square root of 2. On this next figure some of these ratios are shown as:

(OA/Oa)=(OB/Ob)=(OC/Oc)=sqr(2)

This is the proof that an universe line can be generated from the square root of 2. These lengths are exhaustively from the ultimate discrete matter to the universe length.

Now, Pythagoras and the modern Mathematics can tell that irrationallity is concerned only for the diagonal in a square of side equal to 1.

Precisely, this statement undelines their great ignorance about the universe Unity Principle.


What is the Unity Principle?

- what + is + the + unity + principle (fpt.vn, Vietnam)

The unity principle allows to define the universe quantities as a coherent system according to any choosen Unit quantity. For example the word Number 1 is not the absolute symbol representing an absolute length quantity. But it is any length defined as a concrete length quantity that can be denoted as the Unit length. For example, a given "table" can be choosen as the "table unit" for all the universe classes of tables. It is so for an apple fruit. A concrete fruit recognized as an apple is then recognized as a Unit fruit for all fruits of classes "apple".
Basically, it is from this principle that in Dakhiometry, any given length AB, can be used to represent all the universe quantities according to continous spatial constructions, exactly point to point. Any point on a segment AB can represent the space discretness of continous and successive quantities by spatially constructions. There is no irrationnal indetermined point on any length AB what ever its length is.

This principle means that the universe is composed by the Uniqueness Principle. According to that, any unique universe object is a Unit. You will understand it more easily with practical examples as following:

According to the Unity Principle we can take any Quantity as a unit of measurement. Thus, from any the following ratios we can choose a length for the unit of measurement. For example from:

(OA/Oa)=(OB/Ob)=(OC/Oc)=sqr(2)

We can state that Oa, Ob and Oc as the unit for measurement. Thus, we have:

(OA/unit Oa)=sqr(2)
(OB/unit Ob)=sqr(2)
(OC/unit Oc)=sqr(2)

Practical exemples of measurement systems:
(OA/1meter)=sqr(2)
(OB/1N)=sqr(2)
(OC/1kg)=sqr(2)

Consequently
Because OA, OB and OC can be any quantity from the elemental one to the whole universe matter quantity, we can see that the lengths OA, OB and OC can be as:

OA = sqr(2)
OB = sqr(2)
OC = sqr(2)

Because any quantity of matter length as OA or OB or OC, is precisely determined then, the square root of 2 is necessarily a rational quantity.



- who + discover + that + circumference + over + "que" + diameter + equals + pi (Colombia)

- is + pi + the + same + value + for + every + circle

For every circle, in a given spatial system of figure, a length Pi is equal to a same quantity, according to the unit choosen.

The modern Algebra BIG-BUG

How to do an error when one tells that in the square the diagonal is irrational with its sides

Convention of writing,

[sqr(-)] means [square root of (-)]

- simple + resume + for + "applay" + new + work (Autralia)

In a square form the diagonal [c] of side [a], the Pythagoras relationship is :

c² = a² + a²       (5-2-1).

In the continuation, by automatism of writing operations and not by reasonning, this equation is the transformed in Algebra as :

c² = 2.a²       (5-2-2).

From this result, one established that the square root of 2 is irrational because the diagonal and a side are uncommensurable together. We see herafter that a such formalism is an great ERROR of reasonning

Note that a clumsy manner of manipulating this equation should be known only with the Dakhiometry Theorem General on Triangle.

The bad manner is situated in the deformation of the Pythagoras' relationship that is not a simple formalism that one can apply applay simplifivation and reduction. Then, equation (5-2-2) is an error that cancels the meaning of the Pythagoras' one. Example is seen in the destruction of the extended form of this latter. Here, the introduction of the term [2] is a fault of introducing an uncoherent datum. This fault is then undertood as the uncommensurable irrationality while it was only clumsy manipulation of numeral.

How to do not deform the Relationship of the Triangle ? It suffice now to understand the Dakhiometry Theorem General on the Triangle. Here, the equation (5-2-1), must only be written according to a coherent variables. One should not take an apple as a pear...

Then to seak the measurement of square diagonal [c], we can write :

c²/c = (a² + a²)/c       (5-2-3).

Thus, the hypotenuse or the squared diagonal is :

c = (a²)/c + (a²)/c       (5-2-3).


Figure 4-5-3
From (5-2-3), the diagonal [c) is directly given by the Dakhiometry LAM-CA theorem as shown on this figure 4-5-3. That is :

a² = c . (c/2)
et :
a²/c = (c/2)

Finally :

c = 2 . (c/2) = c

- square with diagonal (Massachussets, USA)

The fact that one draws this circle diameter (c) and then, following the square (a2), proves that the circle diameter as square diagonal is directly commensurable with the square edge! This is due to the relationship structure property of the absolute space.
Therefore, the Pythagore's and Socrate's proof as reported by Plato's description and also to the modern Numbers Theory. all these irrationality demonstrations are false because the absolute space is ignored.

- 5 + pains + et + deux + poissons (direcpc.com, USA)

- Mathematique moderne classe de 2ième en France (Cambodia)

- mathmatic (Iraq)

- mathematic (Nigeria)

- mathematic (Turkey)

- "mathmitic" (USA)

- mathematic (Mongolia)

- mathematic (visky.vn, Viet Nam)

- mathematic (alfanett.no, Norway)

- Mathematic (hinet.net, Taiwan)

- mathematic (surfer.at, Austria)

- mathamatics (sasol.com, South Africa)

- mathematic (Hungary)

- mathematic (Poland)

- mathematic (Albania)

- mathematic (Azerbaidjan)

- mathematic (Latvia)

- mathematic (Iceland)

- mathematic (Thailand)

- mathematic (Caucasus)

- mathematic (Indonesia)

- mathematic (shabakah, Saudi Abaria)

- mathematic (telkom.net.id, Indonesia)

- mathamatics (Sri Lanka)

- mathematic (mng.net, ULAANBAATAR, Mongolia)

- mathematic (Georgia)

- mathematic (dion.ne.jp, Japan)

- mathematic (Romania)

- MATHEMATIC (Mexico)

- mathematic (China)

- Mathematic (Malaysia)

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The Thales as like the Pytagore's theorem are only empirical relationship that rest on no reason then their practical verifying and recognizing practical applications. It is thus for most of Mathematics theorem that are nothing than from empirical method.
The true proportionality rest on more basic reason. This is used and rationally proved in the Dakhiometry that rest on the basic property of the ansolute Space. Even if one used to see and practise the Thales theorem, it is more fertile to know what are the reason of Proportionality. The Thales theorem is only the practical consequence of the space law that one should get in conscience to have a more reasonnable and fertile Science.


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There is really a practical, simple and complete method of calculation of space dimensions with the usual four basic operations as addition, substraction, multiplication and division. It can perform as it is done in the current algebraic mode for any complexe calculation and for any root.

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- did + they + use + pi + to + figure + out + that + the + earth + was + round

OK. For everubody, the Earth should be ROUND !

- "eureuka"

About the irrationality of a diagonal relatively to the side of a square, proved by Socrates-Plato

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According to the Homotheic property of linear space, a ratio of (diagnal/side) is always true because any circle from zero to unlimited side length is always rationally defined.

Let (d) a diagonale and (c) the side of any square. A little more complicate and because one know only the empirical Pythagoras theorem, it was used to establish the diagonal and side relationship as:
d²=2.c²
From there it was concluded that there is Irrational Number.
This result was only systematic error or only ideological consideration. Because the relationship {d ; c} is simply a rational ratio as:

(d/c) = Square root2(2)

The relationship between d and c is rationally found as follow :

Let (n_i) any length of a square side and AB_i its diagonal. According to Thales, we have:
AB_i = n_i.sqr(2)

Theorem:
In any square, a diagonal is directly and precisely proportional to its side and is spatialy constructible.


In Dakhiometry construction of multiplication and division are known. It can be constructed
(AB_i/n_i)

It can be find for any square the same length constant as:

AB_i/n_i = sqr(2) = Constante

"- what + is + the + laws + that + a + pi + can't + do ?"

...... :-)


- mathematic + concepts + - + square + of + a + square + root + theory (Canada)

Here is a a deep understanding of space that drive to fertile opening in mind.

One basic particularity of space that allows us to go through space knowledge, is what allows us to get the alphabet with what space construction way can follow.
Pecisely any elemantal fact of space is always a source. For example, knowledge beging on the formation of a square, like a dream that starts to transform itself in the reallity. Thus the elemental space, is alway the anterior of any construction of logic. The elemental space should be seen as the root of any measurement. It is why the basic space theorem is the Lamca theorem that state: any unit of space object appears in action with its squared quantity. There is not any fixe length or any fixed number. Thus there is no fixe unity. The unity is only the starting of a particular system because the most important law is the recurrence fact if matter. It why there are motion. From there energy as the square of motion, is the most important physic factor in the world construction. Because there is no fixed unity, the belief of a square or a circle unity are only human dogma. Most of us are educated to see the world as an eternal coagulated thing. Until modern Physics that considers the world as a geometrical fixe space standing in any time differential.
The matter consruction is always running forward. It is why in space, it is difficult to know everything backward. Thus, you can esealy make addition or multiplication of lengths, addition of angles but these proces reciprocal seem particularly impossible. It is thus the angle trisection that is the dividing of angle was never resolved until now. Because space is not conventional human language one can't lie to himself as it is practised in Numbers which is only conventional human languages. They are not universal truths. So everybody complains about irrationnality of numbers forgotten that it is not numbers that are wrong but because they are only human's imperfect representative language conventions.

In brief, "square of a square root " is really an opening into the deep space knowledge, the fundamental one.

Here is the basic understanging of numbers that are only a means of language to represent the matter recurrence property. It is why only series of numbers are authentic representative concept of the matter recurrence. The particular on some numbers, as transcendental or irrationnal ones are pure human contradictions.

Consider the natural number series.
- APPLICATION OF NUMVER (UK)

Let (n) and (x) two integers. Consider that (n) is the greatest of the series and (x) is any of the integer that may constitutes an unity as reference. We know that n is unlimited but a countable number. The natural number numver series can thus written in such manner that "square of a square root " is the most important understanding of the Recurrence. In the following how to make series of squares numbers from the natural numbers one.

We have starting:

nx
But
nx = (n-x)x = nx-x²
Then,
nx = (n-x)x-x² = nx-2x²
In continuation
nx = nx-3x²
nx = nx-4x²

Finally from the integer multiplication series one can get corresponding unlimited square series of integers as following:
nx = nx - nx²

But the recurrence is a permanent matter property. Therefore we can conclude that there is an unlimited series of square numbers.
They are resumed in this equation:

nx = nx - n(nx²)

Conclusion:
The above formula shows the Unbounded Concept. Unbounded starts from the beginning of a series and contues unlimited. While the infinity concept is a bad and contradictory conventional process that defines a precise particular object: the "end". The contradiction is that one can't define an "end" that is not defined because it can never be reached. So the infinity concept is a tautology.

Starting from unity elements, the squared numbers represent the limitless of the N series. This is done by the limitless of (n) and (x) as we can see in this latter equality. This latter doesn't represent numbers but only the Recurrent matter property. There is no infinity in this representation which is a false dogma. The meaning of this idea rests only on the Permanent Fact of matter.

Another important conclusion from these integer extractions. Because (x) is a Whole Number:
- The + square + root + of + any + whole + number + is + a + rational + number (Canada)

And the numerals that are least than 1 are also rational, because in space language there are precise constructions to build any numeral between Zero and One as well as any one in the unlimited series N.

The Euclide Division algorithm is only an as-like-it procedure, that generates errors, the so called irrational numbers. The so called square root of a Unit Square is pure fantasy of the ignorance because in Geometry, it is a logic contradiction to telle what is an absolute Unit Length.! Numerals are only relative it is why they are exclusively in series: the Recurrent property.

- what + is + the + diffrence + is + it + between + drawing + a + geometric + figure + and + constructing + a + geometric + figure (Canada)

You are right to tell precisely this difference.
A geometrical figure is only a drawing style as for example the Cubism Art. While one is constructing or reasonning logically with space elements as lines and curves. The purpose is to reveal finally some truth behind these logical reasons combinations. And the producing figures is reason constructions that is an absolute proof of universal true spatial language.

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Fig-5
This figure-5 resumes the practical uses of the circles and square equalities. On can obtain any circle or square, from an initial squaring a circle. This needs only the use of the uniqueness of angular characteristic of the equality figure.


- UNIQUENESS OF MALAYSIA (Malaysia)