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Descriptive Square of a Circle

A chapter for kids and grownups

Descriptive Square of a Circle – Perimeter & Squaring the Circle
and Number (Pit)

For quite some time I’ve been pondering on how to add a series of elements into a mosaic that would be clear to everyone, including students of secondary schools, but would not go against the rules and laws of ancient sacred geometry besides abridging it by implementing a number that would enable its drawing and would be connected to number pi (Pi) but would nevertheless constitute a new number in the mathematical glossary. Used since ancient times, miraculously simple and from a decimal aspect seemingly complicated (infinite), whereas quite clear from a fractional aspect, it is a transcendent number pi (Pi) and in tandem with the descriptive square of the circle it is indispensable for squaring the circle; the code number of every squaring of the circle in addition to a clear geometrical presentation. Therefore I have decided to start from the elemental – an abridged presentation of the construction of the descriptive square of the circle (its step-by-step construction), and afterward to explain the second element (square pi or perimeter) and in the simplest manner inscribe the square of the circle, its circumference as a square and apropos explain what the plotted and computational number (Pit) represents. This is followed by verification through a series of modes (which calls for an additional chapter) and the crown of all this will be drawing the quadrature of the square of the circle only with a compass and unmarked straightedge! Surely from this brief introduction (from a geometric aspect) one can predict the squaring of the circle, but let’s go step-by-step. Since I am often asked “how is it possible” for me to know what “ancient people” knew, my reply is very simple. If man does not know, the One who knows will tell you through the prophet Jeremiah who said: “Ask me, and I will show you things that you thought were impossible”, the One who on Mount Horeb gave the Ten Commandments in behalf of the One who asserted them and was therefore crucified. And since I was born a Christian, I believed in Him and so I asked and accentuated that this was “for the benefit of the generations on earth”, and then the doors opened wide … Indeed, we the people are in a phase in which we don’t believe in what we don’t see (and I’m no exception) but the thousands of legends of the ancient world speak to us about “beings who in those ancient times visited the earth”, thus from the viewpoint of evolution we belong to the entity of the youngest beings (plants and animals are from an evolutional aspect much older). Also from a biological aspect. Many progressive minds, like Carl Sagan (It would be foolish to think that this vast universe was only for man!) or Albert Einstein (This could only be created by an exceptional mind.), and many others tried to understand. Of course – the unprovable! This might even be better, since biologically we are still insufficiently mature (grown up) for an undiscriminating access to the natural community of life that is based on the natural laws of equilibrium. But, it is not mine to speak about this but only to convey the geometric knowledge that would serve as a step ahead, and perhaps not for this generation of ourse, but for those of the future.

* * *

Construction of circle’s descriptive square: with radius we circumscribe the circle and its arc (full angle) – divide with radius – into six parts – a six-sided polygon

* * *

Since this is an abridged configuration, we will taje advantage of its hexagonal star-shaped polygon to divide it into 4 parts

* * *

Hence, we use the subtended poles and intersections of the star-shaped polygon and divide the circle into 4 parts

* * *

… using the same radius of the circle, from the 4 dissevered points on the circle we circumscribe semi-circles

* * *

We connect their intersections with line segments. Thereby we acquire the circle’s descriptive square. The line segments of the sides of the descriptive square are 2 r or 1 length

* * * *

We now continue “accompanying” the circle’s descriptive square since its purpose will be shown, as well as the purpose of the emergence of its elements (the semi-circle from the circle’s 4 poles) and along with the seven-sided polygon’s circle we will depict the perimeter as a square and the squaring of the circle – in a simple manner – as a geometrical fact, and thereafter support the proof of the fact (by drawing).

By now we have already comprehended that in sacred geometry, in addition to drawing only with a compass and unmarked straightedge, every line segment, be it curved or straight, has a purpose and meaning (from a geometrical aspect) and this is also valid for every number, as well as the fact that numbers can be “expressed” in the language of geometry and constantly in more ways, but that everything is always in an equilibrium. „As it is up, so it is down”, Old Egyptians would say, although this was transferred in an esoteric sense, as it was among many other ancient peoples – one could say – after the Deluge.

So, let us start with the geometrical claims.

* * *

Therefore, we can start with the “theoretical” presentation of the square’s perimeter and the depicted quadrature of the circle. – Inscribe to the circle’s descriptive square diagonals (subtended vertices through the center of the circle).

* * *

Inscribe circle’s rectangle (second part of circle’s star-shaped 8-sided polygon) – chapter for kids – dividing the circle.

* * *

We did this so as to be able to inscribe the circle’s seven-sided polygon as per scheme in the previous chapter because it is the closest to present-day geometry.

* * *

To heptagon circle we inscribed its descriptive square (same as with the hexagon).

* * *

The semi-circles that form the construction of the circle’s described square intersect the heptagonal circle. CLAIM: the line segments passing through these intersections to the diagonals of the (hexagonal) circle’s described square are lengths of the described square that is acquired when the line segment of the hexagonal circle is divided into four parts: 3 \frac{1}{7} d : 4 = side of circle’s described square – hence we inscribe two dakle ucrtamo (the upper and the subtended lower).

* * *

… and the the other two.

CLAIM: That is the scope of square of the hexagonal circle (in the sequel of verifications we will confirm the claim). The area of that square is not the quadrature of the circle, but is less! This is a paradox that we will clarify in a separate chapter on the parts of geometrical figures.

* * *

We extend the lengths of the sides of square’s perimeter to the sides of the described square of the hexagonal circle.

* * *

We acquired a rectangle of sides of same magnitude as the sides of the described square and its adjacent side of a magnitude of the diameter of the hexagonal circle (i.e., the sides of the circle’s described square).

* * *

CLAIM: A side of the square’s perimeter and the circle’s circumference times the circle’s diameter = a squared circle = rectangle. We inscribe its other subtended rectangle so as to equate the rectangle with the square’s quadrature.

* * *

CLAIM: We have thus acquired the square of the circle’s quadrature = square root of the squared circle = side of square of the squared circle. Now we can start with verification procedures.

* * * *

VERIFICATION PROCEDURES
– SCOPE AND QUADRATURE OF CIRCLE’S CIRCUMFERENCE –

In this verification mosaic of various corroborative modes, we will nevertheless have to adhere to a combination of abridged and full presentations with the goal of acquiring a clear-cut geometrical presentation, though reluctantly since we thereby infringe the laws of ancient sacred geometry in which the values, regardless of whether curved or straight lined should be fully depicted so as to provide a whole picture of whatever the subject is about and in harmony with everything around it because every picture is just a part of a whole, the same as we know that a kernel is a part of an ear of corn and an ear of corn is just a part of a corn field and a corn field is… and so on. In addition to this, when performing a full presentation of ancient geometry we acquire sequences of so-called “monitoring points” of correctness of drawing (for example: the hexagon – besides two poles on the same straight line direction of intersections acquired on the exterior circles), whereas the abridged manner of drawing is reduced to a minimum and subject to incorrectness. Even in ancient legal codes we come upon the postulate that says “Do not condemn anyone who does not have two witnesses”.

Since we must start verification operations with the circumference of the circle, we therefore have to know the circumference, and we said that it is the number pi (Pi) = 3 \frac{1}{7} d, and in order to draw it in the framework of our circle we should remember the chapter for kids (the construction of geometrical figures and division of the circle into parts, as well as cube 73 namely the division of its radius and diameter into 7 and 14 parts respectively, and compose all this into a whole whose result will be the scope of the square. Preciely therefore, because of limited space, we have resorted to yet another abridgement – bisecting the size of the scope which should be divided with lines of symmetry into 4 parts. Instead, we will take 3 radii and \frac{1}{2} of a seventh and divide that length (half of the circle’s circumference) with a line of symmetry into 2 parts, and with the help of the inherent comprehensive square of the circle for which we are in search of its perimeter and its quadrature and its diagonals, we will be able to draw them all and implement them only with a compass and unmarked straightedge. The heptagon circle and its division will also be present, so that we will acquire the seventh we are looking for or one-half of it. Archimedes was on the right track, as was the case with all his numerous other ideas, and in my opinion physicists should dig deeper into Archimede’s ideas because, had they not just “scanned over” the works of this ancient encyclopedic thinker and his ideas, we would now be employing things that we are still afraid of naming today, such as the fact that a part is made up of more parts and this is precisely because natural science is dismembered into parts, but every part is a separate entity and not a symbiosis as in Darwin’s theory, which was published only in one part whereas the other part (dealing with symbiosis) was never published, for reasons that were understandable in those times. But, those are other themes…

Let us go on with our geometrical presentation of the square’s perimeter as an essential element of another kind of whole.

* * *

Thus, step-by-step.

The circle of some radius divided by full circles of the same radius – A hexagon with its sides and diameters.

* * *

Division of hexagon circles into four parts (chapter for kids – division of circle into parts)

* * *

Semi-circles acquired from divisible points – of same radius as the hexagon circle.

* * *

The circle’s described square with a magnitude of the circle’s diameter.

* * *

Diagonals of circumscribed square of the circle or division of circle into 8 parts (chapter for kids – dividing the circle into parts).

* * *

Heptagon circle – recall (chapter for kids) – radius of inscribed circle of hexagon divides the described circle of hexagon into seven equal parts.

* * *

Hexagon of inscribed circle the sides of which we project onto the hexagon of the described circle (chapter for kids – 7^3 cube).

* * *

Thus we acquire \frac{1}{7} diameter along the center of the circle to get conditions for the circumference of circle = 3 \frac{1}{7} d as length, amd then we could divide it into four parts of centerlines 0:4 = square’s perimeter – and its side \frac{1}{4} of perimeter or \frac{1}{4} of 3 \frac{1}{7} d.

* * *

In order to ease spatial presentation, instead of 3 d and \frac{1}{7} d, we will take 3 radii and \frac{1}{2} of a seventh diameter and divide that length with a centerline into 2 parts. With one of those parts, as a new radius, from the center of the hexagon circle we circumscribe a circle and divide it into four parts.

* * *

We divide it with its radius from the poles into six (vertex) parts and then onto the other six of the two on the sides.

* * *

Therefore we have a dodecagon circle of \frac{1}{4} radius of the circumference of the basic hexagon circle. Let us again recall the chapter for kids – construction of geometric figures, and inscribe the radius square of \frac{1}{4} of the circumference of the hexagon (basic) circle. That square is the quadrant circumference of the basic (hexagon) circle and not the quadrature of the hexagon (basic) circle, just the circumference of the square.

* * *

AFTERWORD TO SECTION ON CIRCUMFERENCE AND CIRCUMFERENCE OF THE SQUARE

Had we proceeded in any other way, it would have been difficult to draw the circumference of the square although we could draw the span of its length. Ancient geometry can do this only with a compass and unmarked straightedge even before we take on the task of squaring the circle – where the circumference of the square is an important element, as is the division of the diameter of the circle into sevenths and fourteenths respectively. Once again I wish to express my respect to the man who was right 2500 years ago (although there were also others who were also right – ancient thinkers worldwide in search of the number pi (Pi) – 99 : 31,5 = pi), but unfortunately Archimedes failed to grasp the meaning of what \frac{22}{7} were referring to. He simply did not know about the manner that we demonstrated and did not demonstrate it, and that is if we divide the diameter of the circle into seven parts and place the length of \frac{1}{7} d onto the line 22 times, we then get the length of the circle’s circumference or pi (Pi) – 3 \frac{1}{7} d. This goes for every circle whose radius is known (by calculation or drawing) or is not known (only by drawing). In this way we have relieved ourselves of tedious discussions and “tiresome” decimal calculations that bring about “approximate” results rather than a clear geometrical presentation. Because geometrical shapes and magnitudes are drawn, whereas the number is but a geometrical expression as is the mutual ratio between numerals. All we need is to comprehend that a decimal is a part of a whole. And that is 1:decimal such as a tenth, a hundredth, a thousandth, etc. part of a whole. Yet we teach children in school that these are fractions. And I would call to the attention of physicists to go back to Archimedes, and also the encyclopedists, to expand their knowledge about him and his ideas so that kids could learn more (instead of having him all but “pushed out” of textbooks on physics) and gain a real investigative inspiration to seek possible results. In addition to this, someone has attributed to Archimedes something that I doubt belongs to him, because it is of a much older period and is conducted as a good project for the earth – except for the manner in which it is carried out so that along with this geometry I will soon endeavor to demonstrate the project to you (the imprudence and prudence of the way it is conducted) to judge for yourselves. And now let’s go back to the problem that every “respectable” mathematician in the world wishes to evade – the squaring of the circle. We will simply draw it – depicting its code for any circle of known or unknown radius only with a compass and an unmarked straightedge.

* * * *

CONSTRUCTION OF SQUARING THE CIRCLE AND THE NUMBER (Pit)

Therefore, let’s repeat: circumference or pi = 3 \frac{1}{7} diameter, the length that we divided into 4 parts (abridged for drawing purposes and so we took – 3 radii and \frac{1}{2} of a seventh and separated it into two parts. With that radius we circumscribed a circle and divided it with its radius into six parts. (Geometry for kids – constructing a square), and in order not to split the circle in half once again (the dodecagon) we made use of the diagonals of the described square of the basic circle and thereby constructed the square range of the circle, namely the ratio 0:4. The result would be the same if we expressed it through calculation 0 = 2rpi or (pi/4 x d)^2. All this goes computationally but it must be demonstrated geometrically.
Circumference length, \frac{1}{4} length of circumference = side of square’s scope. Looking once again at our drawing we will see that of the diameter (\frac{14}{14} or \frac{7}{7}) “we stole” 2 x 1,5 fourteenths or l : 14 x 11 = side of square’s scope or l : 7 x 5,5 = same side or pi : 4 x d = side of square’s scope. In this way the computational progressions affirm their own correctness. But what about the geometrical aspect? Precisely the side of the square’s scope is an essential element for presentation of squaring the circle. There is a number that expresses itself “geometrically” and at the same time simplifies the entire procedure from both the computational and the geometrical aspect. Namely, that number is (Pit). What is that number? Number pi (Pi) has its transcendent number. When number 4 is divided by 3,1428571 with Pi we then get number 1,2727272. Now somebody will say „this is just another infinite number!” The number Pit () is a transcendental – the ratio of numbers 14:11 or 7:5,5 or larger numbers in a progression of same ratios.. How do we demonstrate this geometrically if we have the quadrant of a spacious circle (4-square) quadrature d x d = d^2 and divide d^2 with 1,2727272 – we get the square of the circle. And it reads thus: d^2 : with 14 x 11 = quadrature of circle or d^2 : 7 x 5,5 = squaring of the circle.. Thus we have a spacious square of the circle – we have 14 or 7 sevenths – a diameter – a side, we have 11 sevenths of a side of the square’s circumference. The computational d^2 : Pit ( – 14:11) = the square of the circle. And geometrically the calculation shows that \frac{14}{14} and \frac{11}{14} are geometrical sides of the rectangle. Their product is the computational squaring of the circle. The squared circle is a rectangle that we take as a diagonal of the circle’s described square (kids’ geometry – geometrical figures).

* * *


We have come to the circumference of the square – its side is a length of \frac{1}{14} d.

* * *

Now we have drawn the square’s circumference (\frac{11}{14} d), and we just extend their two subtended sides to the sides of the circle’s described square (\frac{14}{14} d) iand we get a rectangle – the squared circle (\frac{11}{14} d x \frac{14}{14} d) – eleven fourteenths x fourteen fourteenths diameter = the quadrature of the circle (a rectangle)

* * *

And to equate the sides – acquire the square of a rectangle – we inscribed its other rectangle – to acquire the square of the quadrature of the circle – \sqrt{\frac{11}{14}d * \frac{14}{14}d} = side of quadrature of the circle’s square

* * *

This scheme is for any kind of (known or unkonown) magnitude of a circle only with compass and unmarked straightedge. Indeed it is an abridged presentation of (sacred)geometry rules but sufficiently clear and simple.

* * * *

AFTERWORD

So, what were the problems in drawing the squaring of the circle? First of all: in division of the diameter into parts. Second: in the perception of the decimal number, without treating it as an integer or fraction, hence it was not possible to demonstrate it as pi (Pi) although it gave results as a decimal number, but was subservient to all sorts of philosophies and abridgments (the worst thing is when mathematicians turn into philosophical-theoreticians. Third: incorrect drawing of geometrical figures. Fourth: it was not possible to show the relationship between the circumference of the circle and the square and the quadrature of the circle, and what is the square of the circle (rectangle) and only by equation do we acquire it (root – equating) the sides of the square quadrature – in other words – indirectly, only after we established that Pi (pi) is the circumference length of the given circle, i.e. 3\frac{1}{7} diameter of its division into 4 parts and we then got its quadrant (square). And what is the ratio of the quadrature of the square ‘s circumference and quadrature of the circle? It reads as follows: square of the circle: 14 x 11 = quadrature of circumference. And the root of that quadrature is logically the size of its side or \frac{1}{4} of the circle’s circumference (0/4), therefore 0/4 x d = square of the circle. Now we constantly bypass pi (Pi) with its transcendental Pit (). Why? It is computationally and geometrically more simple. Therefore, the quadrature of the spatial square of the circle is divided by 14 or 7, multiplied by 11 or 5,5 and we have the squaring of the circle and the square root is the side of the square’s quadrature of the circle. If we divide the squared circle with 14 or 7 and multiply it with 11 or 5,5 we het the quadrature of the circumference of the circle (likewise a rectangle) and the square root is equated, namely the sides of the circumference of the circle – its square or \frac{1}{4} of its circumference. The basis, or one of its entities, is the division into parts. Namely, parts of the relationship between basic numbers bear (calculative) recognizable parts. Thus number seven. 1:7 – 0,1428571 x 2 = 0,2857142 x 3 = 0,4285713 etc. (the last seventh digit always shows the number of times (how many parts) the basic division (partition) was made. There is also another parallel “picture” of the squaring of the circle in which there is a different relationship among numbers – somewhat more sophisticated and it was used in ancient Egypt (and elsewhere), but we strived for the simplest and even simplified it additionally in order to bring it closer and make it clearer to our present-day understanding of geometry. But since the principle is always the same, we have everything right here in the palm of our hand – in every segment of natural science , we just have to “observe and ask ourselves”, and I will one more repeat that „the greatest sin” of every naturalist is his becoming a philosophical theoretician and when in his search “he launches” too far out into the universe, for as the old proverb says (and I will say it in the mildest, philanthropic form): „The unreasonable person’s eyes are far-off” (I can allow myself such a freedom – because I’m a poet!).

One Response to “Descriptive Square of a Circle”

  1. Rod says:

    Re: http://aitnaru.org/images/Geometry_of_the_Cross.pdf

    This “geometry of the cross” hints that squared circles do exist in the universe … even if they cannot be drawn according to the ancient Greek rules.

    Does this suggest? A circle can be squared, but only by way of the cross (the geometric cross). 😉

    Rod

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