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Tle Last Star (division of a circle – educational)

CHAPTER 24

THE LAST STAR
(DIVISION OF A CIRCLE – EDUCATIONAL)

In the last chapter on the pyramid I owed my readers an explanation, namely, if the pyramid was composed of four equilateral triangles, its height would then be divisible by 10 times 7, which signifies a “declivity” of equilateral triangles to the inclined plane of 57.142857° or to 400 sevenths, which is an indication of the relationship of numerical gradient values of 400° divided by 360° = 1.11111′ or of the numerical ratio 10:9. In any case, I just had to say that I’m extremely glad when I have the opportunity to do simple educational programs in geometry for youngsters that are performed only with a compass and unmarked straightedge. This “last star” basically renders such an opportunity. Anyway. Isn’t it much nicer and better when things are simpler? However, in the world, or at least in my homeland, there is a tag line that says: Why simple, when it can be complicated? Others may say, in reference to a relatively elderly man like me: There, he’s becoming senile! It could be, since it is even generally approved as normal, but precisely because of that I want to go back to the beginning, to the pure simplicity of partitioning a circle only with a compass and unmarked straightedge, the basis so simple that a child can comprehend and just as useful for adults, step by step and a series of drawing after drawing! I’m sincerely glad because of this because this method does not leave any mysterious “leftovers” behind, enigmas from which our contemporary world has unfortunately emerged, a world that detests the term simplicity and substitutes it with the term primitivism. However, in order not to complicate and philosophize, let’s turn our attention to this educational chapter – division of a circle into parts by applying the base: the hexagonal circle and its star-like straight-lined hexagon.

* * *2401

Already by dividing the circle’s arc with its radius we have divided the circle into six parts.

* * *2402

The line segment from peak pole to its opposite pole divides the circle into two parts (length from pole to pole – two radii – diameter)

* * *2403

With radial lengths of the central circle every other pole of the hexagonal division divides the circle into three parts. All this is quite clear from our former study of geometry.

* * *2404

Since this geometry with a compass and straightedge is completely inside the circular arc, the hexagonal star-like polygon comes in handy. Every other pole depicted in straight lines.

* * *2405

How does this help in the division of circle into equal parts? It is apparent: with two diameters, pole and opposite pole, as well as intersections of the star-like polygon. Four parts …

* * *2406

… and 6 diameters. The circle is divided into 12 equal parts. Now we take a step further, but in a somewhat different way.

* * *2407

It seems as if we have neglected the floral pattern coupled with the rectilinear star-like pattern. The opposite pole’s intersection of the star-like intersection and its opposing pole and the diameter of opposed poles – the division into four parts.

* * *2408

Now with the same radius, from those new horizontal poles and diameters through the intersections – we have 8 parts. At this point you will notice that this could have been done in a simpler way – 8 without semicircles.

* * *2409

Hence, intersections of the rectilinear star-like pattern and the “floral”. A division into eight parts (we omit the side poles of the hexagon).

* * *2410

In spite of that, the radius of the opposite pole and intersection of the rectilinear star-like pattern and “floral” is useful, since it “speaks” additionally about all six poles.

* * *2411

Once more a coupling of poles of the hexagon divided into 12 parts. For what purpose?

* * *2412

There is a good reason. Keep an eye on the intersections of the star-like hexagon and spherical manner in which we constructed 4 (we will clarify this).

* * *2413

If we take the radius of the peak pole into the span of the compass, it divides the circle into 5 parts. From the opposite pole of the hexagon the division is into 10 parts. From the three poles of the hexagon this makes 15, from the fourth it makes 20 (25 is somewhat specific because it is the square root of number 5). From the sixth pole the division is 30, from the eighth pole it is 40, etc.

* * *2414

Otherwise, in the case of polygons of negative whole numbers the pole of the polygon through the center to the opposite part of the circular arc duplicates the polygon.

* * *2415

We have already written much about number 7 and its division in chapter 21 of the First Book. We will only repeat this. The radius of the triangle half of the star-like hexagonal polygon …

* * *2416

… divides the circular arc into 7 parts (from one of the poles of the hexagon, and thereof from the multitude of poles of the sequential order of 7 we get 14, 21. 28, 35, 42, etc., with the exception of 49 since it is the square root of number 7)

* * *2417

We now go back from the beginning to four and thereafter to eight diameters.

* * *2418

Thereafter, to the elliptical manner of eight, the first intersections, and describe the constellations.

* * *2419

That radius divides the hexagon circle into nine parts and this conforms to the radius in which eight divides the triangular base of the triangle.

* * *2420

This is the division of the circle into nine equal parts and from the opposite pole into eighteen, and so on in sequential order.

* * *2421

From the circle partitioned in tenths we detected how simple it is to solve through duplication of number 5, and now let’s go back to number 4 in the spherical and rectilinear mode since the radius of number 11 is located here.

* * *2422

Number or division into 11 parts only from the peak pole of the hexagonal circle, therefore is multiplied by (2, 3, 4, 5, 6, 7, 8, 9, 10) from which the sequential order of number 11 is acquired.

* * *2423

We have the division of 12. What about 13? It is one of the numbers that are always difficult to divide, but it is still possible when linked to numbers 4, 7, 11. This gives birth to its various star-like polygons. In this case we have decided on the joint intersections of 6 floral and star-like patterns with 4 spherical ones inside the circle of the spherical square.

* * *2424

These intersections reflect the line segments to the hexagonal circle and then we take the peak pole line segments into the compass to the points on the circle and from the same peak pole we divide the circular arc unti the compass comes back to the starting point. Now we have divided the circle with a thirteen-sided polygon.

AFTERWORD

Divisions 14, 15, 16 and other sequential orders are logically simple. Sequence 17 has already been described in the “Last Star” chapters, whereas 19 and 23 more frequently occur as products of various angle divisions, but perhaps this is a little too much for child education purposes since 11 is somehow fair to middling but 13 is much harder and therefore it is no wonder that number 13 is avoided and is the subject of spins and all sorts of stories. Anyway, once time used to be calculated by the moon, from new moon to new moon, and on each thirteenth lunar month was a cyclic-spiral movement of time in accordancew with the path of the spiral galaxy until the time machines – sundials – came into use. However this was not in accord with the Word of God and it has remained so to this very day! What is even worse, this has been transferred to the soil and the people. It has clearly been put into words “Every seventh year do not sow seeds over the ground or harvest a crop”. Why? So the microorganisms in the ground may rest. Today there is no adherence to this rule. Quite the opposite. Of course the soil will be exhausted by such exploitation regardless of the new technologies and afterward long-term land recovery is required. The same goes for people. It is like folks used to say: “Nowadays all days are workdays”! Therefore the generations should not be surprised that things are as they are. But, who knows, maybe I’m still an optimist and my very hopes lie in the belief that generations of the forthcoming civilization will become aware with the decaying of this generation. And it seems that this has happened to earlier civilizations as well, because history repeats itself.

Croatia – Rijeka, June 5, 2013
Author: Tomo Perisha
Webmaster: SLIM
Manuscript transcription + English translation: S.F. Drenovac

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