Direct formula for Pi: Pythagorean composition.
The squaring Pi, as exponentiable number
Of ferman (2009-Aug.): Fernando Mancebo Rodriguez--- Personal page.
Introduction: The squaring Pi is the Pi number easily obtained by direct functions of the circumference relative parameters with application of the Pythagoras Theorem: Number Pi that has important mathematical and geometrical qualities, as well as, squaring connections to multiple levels, that is, all those qualities that the Pi number should have.
Formula by the circumscribed square P=8
Exponentiable or powering property of the squaring Pi means:
Algorithmic Pi versus Squaring Pi
Inadequately, algorithmic ways measure Pi as it was a straight line, by means of cutting portions of circumference and gluing them in straight line (i.e. addition of polygons sides) or similar procedure as series.
While the squaring Pi is measured in its natural curved line.
On the other hand, why would the series of numbers give us Pi, if no circumference parameter is used?
Leibniz : 1- (1/3) + (1/5) - (1/7) + (1/9) ...... = Pi/4
Wallis : (2/1)*(2/3)*(4/3)*(4/5)*(6/5)*(6/7) .... = Pi/2
Nevertheless, and as we can see in the drawing inside the Cartesian coordinates, power functions can define curves, and vice versa, many simple curves can be represented by power functions.Initial idea (about 1980):
So the algorithms ways are alone methods of approximation that don't give us the exact expression of Pi; instead, the correct mathematical expression is made by the Squaring Pi, (which are based in functions of the parameters that build the circumference)
Thus algorithms go extending the curved circumference line to convert it into a straight line, and in this position is measured, while the squaring pi measure the circumference in its natural curved line.
Philosophy of Pi:
The number Pi can't be a transcendental number, but a simple and easy to obtain number, also function of the inscribed and circumscribed squares to the circumference, as well of its diameter, like it is seen geometrically.
That is, as any other simple geometric figure, the circumference (or Pi) can be obtained by direct functions of its building parameters, as can be its diameter.
* Exponential number (not transcendental) Pi^n
* Direct function of the diameter of the circumference (r=1). (below)
* Direct function of the inscribed and circumscribed squares to the circumference. (drawing 3-1 )
* Function of integration among Squaring Pi - circumscribed square - and - decimal system (10) -anterior drawing-
* Its powers square with diverse inscribed and circumscribed circumferences and squares among them.
* Floating point structure and adjustment by powers.
You can see many of my works, in the following pages:
Double slit and camera obscura experiments: ferman experiment
In favour of the cosmos theory of ferman FCM ||| Theory of Everything: summary
Model of Cosmos. ||| Atomic model ||| Development speed of forces.||| Magnets: N-S magnetic polarity.
Stellar molecules ||| Static and Dynamic chaos||| Inversion or Left-right proof
Chart of atomic measures||| The main foundations of the Cosmos' Structure
Positive electric charges reside in orbits.||| Mathematical cosmic model based on Pi.
Inexactness principle in observations ||| Einstein and the gravity ||| The Universal Motion ||| Atomic particles
Cosmic Geometry ||| Bipolar electronic: semiconductors ||| Multiverse or multi-worlds
Radial coordinates.||| Physical and mathematical sets theory. | Algebraic product of sets.
Planar angles: Trimetry.||| Fractions: natural portions.||| Cosmic spiral
Equivalence and commutive property of division. ||| Concepts and Numbers. ||| Bend coefficient of curves ||| Mathematical dimensions
Transposition property ||| Accumulated product: Powers ||| Dimensional Geometry: Reversibility
Priority Rule in powers and roots ||| The decimal counter ||| Paradoxes in mathematics
Direct formula for Pi: The Squaring Pi. ||| The pyramids of Squaring Pi. ||| Functions of Pi ||| Integration formulas Pi.
Squaring the Circle
Spherical molecules. ||| Genetic Heredity. ||| Metaphysics: Spanish only. ||| Brain and Consciousness. ||| Type of Genes T and D
Certainty Principle ||| From the Schrodinger cat to the Ferman's birds ||| The meaning of Dreams
Freely economy ||| Theoricles of Alexandria ||| Satire on the Quantum Mechanics
Andalusian Roof Tile. ||| Rotary Engine. ||| Water motors: Vaporization engines.
Triangular ferman's Houses .||| The fringed forest
Garbage Triangle: Quantum mechanics, Relativity, Standard theory
Fables and tales of the relativists clocks.||| Nuclei of galaxies.||| Particles accelerators.
Hydrocarbons, water and vital principles on the Earth. ||| Cosmos formula : Metaphysics
Ubiquity Principle of set.||| Positive electric charges reside in orbits.
Chaos Fecundity. Symbiosis: from the Chaos to the Evolution.||| Speed-Chords in galaxies.
Who is God
To begin, let me make a simple summary in comic form to introduce the meaning and foundation of the Squaring Pi.
Inadequately, algorithmic ways measure Pi as it was a straight line, by means of cutting portions of circumference and gluing them in straight line.
While the squaring Pi is measured in its natural curved line.
"" As we can see if we treat of making the circumference on the points centers, then the circumference must to be situated in the exterior of the union of the circumference points, and this way more large than the real one.
And this circumstance is given for any dimension of the points, already they are infinitesimal.
So, what give us the true radius of the circumference are the union points, but not its center. ""
Algorithms method for Pi sum sides of polygons (A) getting bigger length than summing arc of circumference B arc.
Structural Principle for curves and straight lines
All and each union among consecutive points (infinitesimal portions) of a curve produces an infinitesimal loss of length regarding to the same union if it were made in straight line.
This is due to in curve lines all their points are nearer among them by the interior of the curve.
Loss in circumference ---- (2,4189 x 10^-6) r.
All and each union among consecutive points (infinitesimal portions) of a curve produces an infinitesimal loss of length regarding to the same union if it were made in straight line.
Simple argumentation (in agreement to current Pi)
A.- The method of sum of polygons-sides inscribed to the circumference and another series used for obtaining the current number Pi have similar and parallel solutions, those which getting to their limits, the number of the infinitesimal portions of line are equal in the resultant straight line than in the initial curve line of circumference.
B.- So, it is correct to argue that the resultant addiction of portions of both lines give us the same length.
Double argumentation (contrary to the current Pi)
1.- Accepting the first consideration of the anterior argumentation (A) it is proposed and considered a second different argumentation.
2.- Straight and curve lines have different geometric and mathematical structure and properties, in such a way that the same quantity of portion of line has more dimension and length when they are extended in straight line than when they are shrink or bended in curve line due to these portions are now nearer among them by the interior of the circumference.
Fractal argumentation, contrary to the current Pi
Many and diverse are the argumentations and proofs contrary to the exactitude of the current Pi number, but to those mathematicians that like the general method for obtaining Pi by means of the addiction of the sides of the inscribed square to the circumference it is possible to offer them, as clear, logical and contrary argumentation, the fractal argumentation on the addition of the vertices of inscribed polygons.
When we inscribed a regular polygon inside a circumference, we use alone one point or dot of the circumference as vertex (v) for building two sides of the polygon. (see below drawing)
In this circumstance, when we sum any pair of sides with the same vertex, we sum this unique vertex (v) two times, one time belonging to each polygon side.
Say, in the adjustment of the circumference length any point (v) alone is summed one time; but in the algorithmic adjustment any vertex point (v) is summed two times.
This way, if for example the inscribed polygon has 4096 sides, then when summing we add 8192 vertices, which means that when we go approaching to the limit of sides we are summing more points or vertices than the circumference really has.
With which, the algorithmic addition of inscribed sides give us more length than the circumference has because we are adding more points or vertex than the existent ones.
Really this circumstance could be considered as a flaw of the process of summation.
The Squaring Pi consists on a function (exponential) of the inscribed and circumscribe squares to the circumference.
The pyramids of squaring Pi are numeric tables developed in pyramid or triangle form, which show us as successive powers of Pi go approaching to successive decimal powers of the inscribed and circumscribed squares to the circumference, to end up coinciding at certain level.
With the values of these levels of coincidence we can obtain the squaring Pi by means of root of these values.
In this second pyramid, it is shown the power Pi^34 in relation with the perimeter of the circumscribed square to the circumference (8) by the decimal powers 10^16.
As we see, the odd powers of squaring Pi drive us to the inscribed square to the circumference, and the even powers drive us to the circumscribed square.
Here we observe as the Pi powers are approximately the double that the decimal powers (x10^n) applied to the perimeters of the squares, and it is due to get any decimal value applied to the sides perimeter is necessary the square of the number Pi (Pi^2 = 9.8696....)
We also observe that the powers of Pi in relation with the squares perimeters are the order of 2n+1 and 2n+2 due to for starting the pyramids of powers we need of +1 or +2 the powers of Pi to get the first term in the powers of the squares' perimeters.
Reasoning the number n of powers
The number of decimal powers n (10^n) that multiply the sides of the inscribed and circumscribe squares to the circumference is the number of powers applied to the triangles legs that form these sides when they are obtained by the Pythagoras theorem.
It seems to be that the coincidence numbers in powers (n=8 and n=16) for the perimeters of the inscribed and circumscribe square to the circumference are produced to this level due to these n-numbers are the numbers of times that we must to multiply the sides (legs) of the triangles to build the perimeters of the squares, as for the Pythagoras theorem.
Say, to form a side of the inscribed square (hypotenuse) it is necessary to elevate any leg to the square, what gives us as result 4 powers of legs for any square-side and 8 powers to the both square-side inscribed to the semi-circumference (Pi)
* For the pyramid of the circumscribed square the result will be double because of here it is not a semi-square, but a complete square.
Direct formula of Pi
Bend coefficient of the circumference.
* Mathematical maxim of squaring Pi. : "If the circumference is built, contained, limited and changed depending on the value of its inscribed squares (inner and outer), and vice versa...... Then, a direct function of the perimeters of these squares that gives us the exact value of Pi ought to exist, and vice versa ..... A direct function of Pi that gives us the value of the perimeters of the inscribed (inner and outer) squares to the circumference also ought to exist."
"The logic and mathematical principles are not consequent neither they could accept that the two more regular figures of the geometry (square and circumference) didn't have a direct function of common structuring when they share out the same elements and construction parameters as they are the diameter of the circumference and the sides of their inscribed squares."
Subsequently I expose a direct formula (/s) for Pi, which, to have some properties and particular characteristics, we will denominate it squaring Pi.
Now well, when this Pi number has its own name, it already indicates us that some difference of value has to be between the Algorithmic Pi (current Pi) and the Squaring Pi.
And of course, this difference exists and takes place beyond the sixth decimal, that is to say, starting from a millionth.
But to understand the process of development of this number better, I will make a brief summary of its history.
1.- I work and study cosmology for about thirty years, and already a long time ago I reached the conclusion that the Pi number is basic in the construction of the cosmic structures.
The Pi number intervenes this way in the valuation of the unit of atomic mass; relationship between atomic mass and atomic radius; measure of the atomic density (density of atoms), etc.
And thinking about it a little, we can get the conclusion that it should be this way because if we contemplate the Cosmos in its essence, we see that to create the systems that we know when they have spherical construction, spiral form, circular motion, etc., here the only existent basic number is Pi, because this number defines and measures the spherical systems.
2.- Soon after of this, I was becoming aware that also the relationship among inferior systems as atoms and superiors ones as stars, all they should be structured by means of Pi, in this case for functions and powers of Pi.
This way the lineal dimensions between atoms and stars are of 6,28 x 10E22, that is to say, the radius of a star is 6,28 x 10E22 times bigger than the radius of a equivalent atom.
3.- And later on, I discovered something interesting for the mathematics: the mentioned Squaring Pi.
I realized that in the powers of Pi there were levels or cycles of coincidences or connection (quadratures) among the powers of Pi and exponential functions of Pi.
But I also could observe that these quadratures didn't coincide exactly with the value of the current Pi, but with a very approximate value.
And to that approximate value of Pi that fulfils the mentioned coincidences is to what I call Squaring Pi.
But, something much more interesting still was observed. Not alone quadratures among exponential functions of the squaring Pi exist, but also quadratures with functions of 2.
But even more, the squaring Pi belonged together, coincide or has quadrature with the decimal powers, for example, 108, 1016 etc.
Well, as all this seems very intricate, let us put some examples and formulas:
A. - We have said that quadratures exist between powers of the squaring Pi and functions of this number.
For example: (drawing)
( 1 ) Pi37 = (2 Pi) 3 x 1016
Here quadratures are given among high powers of Pi with functions of Pi and decimal powers.
And this property or coincidence is not given with the current number Pi.
B. - Also we have said that the power of the squaring Pi also makes quadratures with functions of 2,
( 1 ) Pi 34 = 8 x 1016
( 2 ) Pi 17 = 2 x root of 2 x 108
Of course, the value of the current Pi neither has these quadratures.
C. - But the squaring Pi gives us something more.
Beside of functions on itself and on functions of 2, the squaring Pi has quadratures with the decimal system, as we can see.
This way in the previous example, we see as beside functions of 2 ( cube of 2, 2 by root of 2 ) the squaring Pi also has quadratures with decimal powers (108, 1016 )
Summarizing, the squaring Pi is an approximate number to the current Pi, whose powers have correspondence and quadrature with its own spherical functions, with functions of 2 and with decimal powers.
Because well, from these last two formulas we extract the value of squaring Pi, which as we see they are not algorithms, but direct formulas.
So, the value of the squaring Pi is:
Now then, and without polemic spirit, I would like to expose some observations that I make myself about the value of the current Pi.
Perhaps I don't know sufficiently the processes of obtaining of Pi, but although this, I ask myself some questions, such as:
Are the methods of obtaining of Pi totally consequent with the geometric reality of Pi, or are they sophisticated and complex systems of series, functions and algorithms that are not adjusted completely to the reality?
Personally, I have always been convinced that the Pi cannot be a number so rare, lonely, hidden, slippery, independent, without connection, etc. but just the opposite.
Pi, as basic element of the Cosmos, of the geometry, mathematics, etc., has to be an index number, open, dependent, connectable; with quadratures to different levels, numbers and functions, and therefore nothing to do with current of Pi.
So I understand that possibly, the squaring Pi is the true value of Pi, since different Pi numbers shouldn't exist.
Anyway, here we have a Pi number (Squaring Pi) that connects at many and repeated levels with own functions, with functions of 2, with decimal powers, and let us hope to discover more connections or quadratures soon.
Next, we can see quadrature of squaring Pi with roots of 8 (2E3) by decimal powers.
Here we can see as the relationship or function among roots (Rn) and N indexes is:
1.- The true Pi number should complete all the geometric and mathematical squaring here exposed.
2.- The true Pi number, as symmetric figure, structured and depending of the inscribed and circumscribed square to the circumference, besides to have total geometric dependence, also it must to have total mathematical dependence defined by means of mathematical formulas and functions of interrelation among the circumference y its inscribed and circumscribed squares.
3.- When not concurring in the current Pi this circumstances, those which the Squaring Pi has, this author consider that the Squaring Pi should be the correct Pi number.
""Well Ferman, very interesting, very promising, very.........
We have the Squaring Pi that is function of the perimeters of the inscribed and bounded squares of the circumference; that is function of 2; that is a very promising exponential number for the development of the cosmological mathematics, etc., but: How you would explain us the numeric difference between that Squaring Pi and the results that we obtain with the algorithms for Pi?"".
Well, I believe that this is explained, and I think it is also demonstrated, by means of the called bend coefficient of the circumference, and of course of any curve.
If we notice the algorithms, let us put as example to the Liu Hui's algorithm, in this algorithm we proceed to sum in lineal form (in straight line) to all the bases of the obtained triangles, while what would proceed would be the sum in curved form applying the bend coefficient that logically has to take the circumference.
As we see in the drawing (c):
If we unite for their sum to all the triangles' bases in that we go dividing the circumference, we see that the result of the union of these triangular bases is a straight line.
But not, what we sought to obtain was a circumference or circle with the sum of these triangles.
Then geometrically it is not the same a thing that the other one.
And what really lacked in this sum.
Of course, the adaptation to the circular form by means of the application of the bend coefficient.
In this sense, if we apply the bend coefficient of the circumference to the results of the algorithmic Pi, we would obtain the real Pi, that is to say, the Squaring Pi.
If for example we have a series of points forming a straight line (Anterior drawing, a - b) and we treat to twist this straight line to form a curve, we see that the points that compose it (b) they change their position and they join in a different way.
This alignment change of the points creates a new geometric form, which will take also accompanied a change of space distribution and of space dimension.
In this sense this author considers that this geometric change takes also accompanied a dimension and measure change, question that could take us to a conclusion or theorem on the transformation of straight lines into curved lines:
"Any straight line that is transformed into a curve suffers a dimensional variation so much for the concave side as for the convex one, appraisable by means of a bend coefficient ( ."
In theory (and from our three-dimensional perspective), if we go applying to any straight line successively bend coefficients with values every time bigger, geometrically we will go bending the straight line and transforming it into a curve, every time with more bend, uniting its tips for a certain value and getting this way a circumference, and later on, if we continue increasing the values of the coefficient, we will be able to go superimposing spires or circumferences some on the other ones until transforming the initial straight line into a single central point.
As we see in the previous drawing, if we bend a straight line for obtaining a curve, in it all and each one of its points is nearer than in the straight line.
This way, in the previous drawing, we see as in the curved line all its points are nearer some from other and their distances (for example b and c) among near points are smaller as the curve is more closed and therefore its coefficient of more bend.
Consequently we see that this dimensional reduction comes given by the same property and characteristic of the space. All this is a spatial consequence.
* Page to see first work about Bend coefficient of the circumference and other curves.
Now well, as in the case of the circumference what interests us is the interior side or concavity, then this degradation would be negative and its dimension would be smaller than the precedent straight line.
Concluding, for me, the algorithms don't give us the exact value of the circumference but the value that this circumference would have transforming it into straight line.
And vice versa: If to the algorithmic value that we give currently to the circumference we apply the coefficient of bend of the same one, we will obtain to the Squaring Pi that this theory proposes.
Therefore, and summarizing, the squaring o powers Pi stops to be a transcendental number to become an index and squaring number in the mathematical structure, and a basic and essential number in the cosmic structuring where it is the basic number to measure, relate and to build the dimensions of the different cosmic elements.
In the previous drawing we show the formula, algebraic exclusively, to obtain the squaring Pi in function of 2.
I would like to revise here the methods for obtaining Pi, always observed from my viewpoints.
To revise the main methods used till now to obtain Pi, we will divide them in three types according to the used basic principles.
These types would be: Convergent, Parallel and Integrated.
Convergent (and orthodox) will be the methods that use of the geometric reality to get the number Pi.
As example, we have the method of Archimedes that starting from the triangulation and use of successive polygons he was getting to come closer more and more to the number Pi.
As we see, this method uses of the geometric reality to obtain Pi, although its difficulty is in the almost infinite number of operations that it is necessary to make to obtain important decimals of Pi.
The Parallel method (heterodox) is not direct method for the strict geometric reality, but rather we use operations with factors that we know they can go near to Pi number, but they are not intimately related with the geometry, but rather they are simple mathematical fractions that we ahead of time already know they will drive us near to Pi.
In this example we have the number series that at the moment we use for obtaining Pi.
* In this sense, it seems to be that the series that try to go coming closer to Pi end up surpassing to this Pi number since in these series their number of factors tend to infinite and they don't have the inflection point that the circumference, sines, cosines, etc. have when finishing their rotational cycles.
This method, also a philosophy of Pi, it is the one that takes notice of the properties of the inscribed and external squares to the circumference for propitiating formulas and operations with powers and roots which should drive to Pi. (See drawing)
* We can remember that roots and powers are the base in the triangulation of squares.(Pythagoras)
This option or method sustains that if the circumference depends and it is built on its inscribed squares, then this circumference should be defined by power and roots of the sides of these squares.
And as in the practice we can observe an apparent coincidence between powers of Pi and the powers of the sides of the squares, because we understand that this method should be correct in its configuration and results for Pi.
So in good mathematical and geometric logic, Pi cannot be a transcendental number without connection by means of direct formulas with its inscribed and bounded squares, but just the opposite; Pi has to be united and defined to its inscribed and bounded squares by means of direct mathematics formulas, and vice versa: That is to say, direct functions of Pi have to give us the value of its inscribed squares; and direct formulas of its inscribed squares have to give us to Pi as numeric solution.
Just what makes the exposed formulas for squaring Pi (so much algebraic as geometric)
In the below formula, we can see as the decimal powers of the square's sides are intersected or crossed with the squaring Pi powers at N= 16, that is to say, where the squaring Pi is built.
To this point, we will call Inflection Point.
This formula points to be a clear proof of the consistence and reality of Squaring Pi as the true Pi
The integration function (that we can see in the drawing) that start from the semi-perimeter of the inscribed square of the circumference, passing by the squaring Pi with value N=16.
Although in this function the most important questions are the elements that compose and integrate it.
These elements are three:
1.- Firstly the value of squared geometry which is the value 8, which corresponds to the value of the longitude of the perimeter of the bounded square on the circumference of radius 1.
2.- In second place, this function includes a parameter of decimal power (10N), which gives to the decimal system the worth of structural geometry.
3.- And in third place, it is connected with the spherical or circular systems by mean of the squaring Pi.
Therefore this function has an absolute integrative property in the fields of the geometry and mathematics in general.
* As we can see, the integration function has also a limit, but in this function we can get its inflection point, which is N=16.
The key moment to conclude for my part that the algorithmic Pi wasn't correct but approximate, it was when working on the powers of this algorithmic Pi and arrived to the power 17, this power gave me 2,82844563.. x 108) coinciding to the two hundred-thousandth with the semi-perimeters of the inscribed square to the circumference, (2,82842712.. x 108)
Similar coincidence could not be given in the mathematical logic.
Then for me the algorithmic Pi was not exact and it was necessary to square it with the semi-perimeter of the inscribed square of the circumference.
Then, the whole work and resolution of this quadrature begin to take place.
The decimal levelling would be a form of mathematical operation in certain circumstances.
This form is also used as way of expression of important numeric values.
For example if we have a high numeric value: 1.234.456.178.225, we can also express it (and commonly it is made in physics) as 1'234456178225 x 10E12.
As we see, here the object or finality if of expressing the total number by means of a single integer cipher, which is followed for many decimals by a decimal power.
However the decimal levelling not alone it is a form of expression of quantities, but a form of operating at different levels where the same norms and operative functions are used.
Let us suppose that stars and atoms have the same structural form, but however some (atoms) are quasi-infinitely smaller that the other ones (stars).
Let us say that the lineal dimensions of atoms are 6'28 x 10E22 times smaller than in stars.
In this case if we are operating for instance with stars' radii and in a given moment we want to move at the atomic level, alone we will have to introduce the factor of decimal levelling 6'28 x 10E22 to be operating with atomic radii.
But you would ask me: Well, but for what all this could serve us? Because due to a similar form is used to obtain the squaring Pi.
Let us see: Squaring Pi is 3'141591444142....
If we elevate it to the square, we will have: 9'86959680 ....
If now we elevate it to the square again we will have: 97'408941 ....
But in this case we have two integer cipher and many other decimals.
To transform these two integer cipher into a unique one, we subject the quantity to the decimal levelling, that is to say, we divide it for 10 obtaining 9'7408941 .... x 10.
And we continue elevating to squaring Pi to the square, given us as result 9'613869728 .... x 10E2.
And so forth.........
Now well, analyzing the results that we go obtaining, we see that when we end up elevating to squaring Pi to the 17 power, an later on to be subjected it to the decimal levelling, the result gives us the value of the semi-perimeter of the inner square inscribed on the circumference of radius 1, that is to say, 2'82842712... (2 by root of 2) x 10E8
And if we continue elevating to squaring Pi until taking to the 34 power, (also levelling), we see that the result coincides with 8, that is to say, the value of the bounded square of the circumference of radius 1.
To show a little the essential of Pi in Cosmology, I put several simple formulas, pick out from my atomic model.
In the following formula we see the Mathematical Unit of Atomic Mass, which also uses the cubic root of 2 when this root coincides nearby with the metric units at atomic level.
Same it will be made later with the Unit of Atomic Radius.
In the following drawing we have the Mathematical Unit of Atomic Radius.
Below, the drawing of the Atomic Radii formula is showed.
Next, we see the formula of the atomic density, that is to say, the density of atoms according to its mass and volume.
And in the following drawing, we have the distances where the orbital of star and atoms are located (Planets and electrons).
This author understands that the Squaring Pi can be the reason and guide in the structuring of the Cosmos through the Fourth Dimension, or exponential dimension of space-time.
And it is so because this Fourth Dimension seems to have a clear parallelism regarding to the property and exponential structure of Pi that before we saw.
Regarding it, let us remember that the Fourth Dimension is an exponential form of structuring of space-time (the structuring of energy and matter) by means of which the cosmic energy is constituted in material points that go accumulating until creating gravitational systems or material units such as atoms, those which in turn unite to form other systems with equal characteristics as they are the stars, which unite in turn among them to form other bigger systems, and so forth.
And vice versa, that the stars are constituted by atoms those which in turn are constituted by sub-atoms and so forth through the Fourth Dimension of the space.
This form is represented by the previous drawing, the Tetacoor. (Tetra-coordinate)
The squaring Pi has many own and particular properties that give it the category of special number.
Perhaps for many year would have discussion about if it is the true number Pi or not, but its particularities and properties will be forever.