free hit counters

Universal code of regular polygons – Children’s education

11. Chapter (III)

Universal code of regular polygons

(Education for children)

My dear readers. As I am a poet (prooved by 100 poems on web, written in one day, just a part of 11000 written in last 10 years), I can tell you thousands of legends, stories much more interesting than this pages. The ones who tries to find something else in them, misterious, from genesis to present times, would come to their theories. But a poem is strange being or someone’s property. Just like Keith Richards from „Rolling Stones“ said 20 years ago (and I did it 30 years ago): „Poem comes by itself and if you don’t write it it goes further, to somebody else:“ If two people from two different areas, not knowing each other, say the same thing, then it must be true. As a poet I don’t dare to write something like that, because geometry is reality, although neglected in elementary school. But it is a bit strange when you find records which could be explained, especially if they are ancient stories, literally transferred for thousands years. Bible, 4.26: „A son was born to Shet, and he named him Enosh. Then a name of Jehova started to be invoked.“ Enosh is fourth in blood line, so we must start with number four, as much as it looks strange. Number 3 is different chapter. It is different from code system which is sequencial. In this chapter you’ll see sequence of whole numbers. The purpose is to construct center of number’s polygon just with compass and ruler without measures. I could call it „tree of knowledge“, but it is just code system of regular polygon’s construction from 4 to next (5, 6, 7, 8, 9, 10, 11, 12, etc.). A poet would say in twelth form and musician in three quarter tact. Don’t take it wrong, a poet in me. Now, a geometry. Ancient, natural, just with compass and ruler without measures (without protractor, which is useless), step by step. Like building a house or planting a tree, preparation of foundation is basic. (again, my story) But it has to be in geometry as well.

* * *


Preparation of foundation. Line and on the line given lenght of some arbitrary size. It will be the base of segment of every regular polygon of regular numbers (regular numbers).

* * *


From the ends of lenght and with the range of compass which is size of the lenght describe simetral semicircles. Simetral divides lenght on two parts.

* * *


With same range from intersection of simetral circles, through which passes lenght’s simetral, describe full circle.

* * *


Lenght’s simetral divides it on two parts. With the same range of compass divide circle on its six parts, starting from intersection of lenght’s simetral and circle.

* * *


Hexagon. Opposite poles connect. Cube (1x1x1) watched from the angle.

* * *


Inscribe in hexagon its star hexagonal polygon. There we have to remember division of cube on cube 3 (3x3x3).

* * *


So, through intersectiones of star polygon vertical next to simetral and by side (left, right) extended on the hexagon’s sides.

* * *


In that way we got points on hexagon’s sides so we can divide cube on cube 3 (3x3x3). That division formed 6 double triangles on simetral (left and right from simetral). Split them too.

* * *


That points of division on simetral, from the ends of given lenght, connect with lines. Those are centers of one segment of regular polygons of 1 – 12.

* * *


If from the circle’s peak pole we describe another circle of the same radius, and divide it on cube 3, we get center of segment of polygon from 1 – 12 on simetral.

What does it mean? If peaks of poygons segment are on simetral, it is simple. Peak of the segment is circle’s center of some ordinal number’s polygon. If we describe circle from every center, and that circle describes foundation (given lenght), the lenght will divide circle on its ordinal number of the center (4 on 4 parts, 5 on 5, 6 on 6, etc.). We’ll realize it now, starting from number 4, and we’ll skip number 3 because it asks for its own chapter. In this chapter we’ll go to number 14, because I think that after ten examples everyone will understand code system of construction of ordinal number’s polygons, if the lenght of foundation of polygon’s segment is given. As I have told you, foundation is very important. And foundation is a pillar of measures made by construction of the cube and its division on 3x3x3 – cube 3, and then on 6x6x6, but just shown on simetral. Therefore, it would be good to study construction of the cube and its division on parts as well as pillars of measures. But let’s start step by step, from number 4, but before that I will add number values of peak’s and foundation’s angle segments of some ordinal number’s polygons. You can use protractor just for check. It has one defect, if you miss for just half degree, polygon is not regular. Another thing is that school protractor is too small and it mastakes. Because of that I am surprised why on the beginning of science about regular polygons is a pentagon whose base angles are 54º, and not nonagon whose base’s angles are 70º (rounded). But we’ll do it without protractor, and that is difference between ancient and nowdays geometry. So, with mesurement rope and straight lath (read: compass and ruler without measures).

The chart. Number of angles (starting from 3). System: 360º: ordinal number = peak’s angle of triangle of number’s segment. Then 180º – peak’s angle divided with 2 = angles on the triangle’s base of segment.


* * *


As we said, start from number 4. Its circles from central point or center of peak angles.

* * *


We take given base in the range of compass. It divides circle on 4 parts.

* * *


Next point on circle’s simetral, its divisions on twelfths. Equitentacle triangle with given base. From that point describe given lenght with circle.

* * *


In the range of compass take given lenght. Divides circle on 5 parts. Pentagon. Peak angles 72º, base angles 54º.

* * *


Next peak. Center. Equitentacle triangle.

* * *


Hexagon. Given lenght’s size.

* * *


Seventh twelfth. Equitentacle triangle. 360º: 7 = 51, 428571º, center of the seven angled circle.

* * *


Given lenght. Divides it on 7 parts.

* * *


Eighth point of twelve part division of diameter or simetral. Center eight (circle always describe given lenght).

* * *


Given lenght divides it on 8 parts. (inscribe just starting segment of polygon)

* * *


Ninth point of twelve part division of diameter. From peak nine describe given lenght with circle.

* * *


Given lenght divides it on 9 parts. Peak (center’s) angles 40º, base angles (of given lenght) 70º .

* * *


Tenth point. 360º: 10 = 36º peak angles. From peak 10 describe given lenght with circle.

* * *


Given lenght divides it on 10 parts.

* * *


Eleventh point. Same procedure. Circle is divided on 11 parts. Peak angles 360º: 11 = 32, 727272º .

* * *


From peak pole 12. 12 – angle.

* * *


To continue sequence from peak pole, we add circle and divide it on twelfths. Next point is 13. Center of 13 – angle.

* * *


Next point is 14. 14 – angle and further would be 15, 16, 17, 18 etc. We could do it in different way. Given lenght and peak pole of polygon’s circle (5 and 10, 6 and 12, 7 and 14, etc.). Duality of the radius, for example: 360º: 28 = 12, 857142º – peak. 180º – 12, 857142 = 167, 14286º: 2 = 83, 57143º .

An afterword

I think you understood the principle. Basic is, in fact, division of a cube on 3x3x3. Cube 3 throughout 2, which manifests on vertical as twelve part division. And those points as peak angles or centers of polygons or descriptive circles of polygon. And given lenght divides it on parts. Given lenght is base of one polygon’s segment. It doesn’t require many words. Geometrical sketch just with compass and ruler without measures speaks for itself. For other pillars of measures you know from last two books (see pillars of measures) but they have different calculation (geometry and numerous) and I’ll show it but without pillars of measures for start. Three is deliberately skipped because it belongs to that system. Have no fear. It is simplier than you think. For adults who are full of dogmas, legends, and for members of different public and secret groups, to please them, I’ll show „tree of knowledge“. So they will know they could ask (and didn’t) him who gave ten commandments on Horeb, more than 3000 years ago, and the one confirmed it 2000 years ago, in more drastic way. I am sorry that no one today didn’t make it easier for you (school), nor Microsoft, nor Apple (leading in informatic technology). They didn’t make educational geometry programms, and compass is also dissapearing from schools (basic tool in geometry). Why do I say it.? With number goes geometrical review and in reverse, and it gives perception of natural sciences. And you are left to hard people’s games. How will harmony come about? In no way. I am sorry for that.


3 Responses to “Universal code of regular polygons – Children’s education”

  1. Ricardo says:

    wow. I did something almost similar to this, your teaching took me to this thinking exactly, you are amazing Mr.Tomo. Thanks again, I was waiting for this. 🙂

  2. Ricardo says:

    All i do is draw these days, and the quadriangle and pentagon have thyr middle a bit lower than you say Mr.Tomo. I drawn it several times, 7,8,9,10 are correct, but 4 and 5 if you draw it like you did in your last chapters and then do the cube of 3, the center is a bit lower… im gonna keep trying, but i cant make the square with the middle point you are giving us here and the pentagon too. the rest came out perfectly like you show. thanks again.

Leave a Reply

Powered by WordPress | Designed by: suv | Thanks to trucks, infiniti suv and toyota suv