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Quadriangle and octogon – Childrens’s education

10. Chapter (III.)

Quadriangle and octogon

(Children’s education)

On the first sight it seems this chapter doesn’t bring anything special about quadriangle, because it can be drawn in many ways, what could be noticed in many chapters of the first and second book oft his geometry. It is very close to normal school drawing, but we saw this ways brought us to the answer of one, so called, most famous enigma; trisection of the angle. At the same time we saw sequences of other products which results from this ancient geometric approach, just with compass and ruler without measures. We are particulary interested in them because they are numerous and so there is no harm in repeating something particulary important, and continuosly repeated in our real world. Especially if we notice one more importance, and that would be transition of quadriangle from concave to convex, border between, we would say, two worlds or dimensions. Because we won’t begin with a point in the space and circle, but with line and some lenght on it. There is nothing mystical in it, it is just different approach to be understadible to every child. Simplified, meaning without drawing full circles, as it is the law of ancient geometry, and just with small number of results of other values which are otherwise hard to get (for example, 13 – angle) in classic way (school) because it is difficult to determine 360º : 3 = 27, 692307º or its duality 55, 384614º, and int his geometry it is not. As 4 and 8 are closely connected, we’ll join them in one entirety with a few „side products“ what will be enough for basic – school age.

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One of the universal principles. Line. On the line some lenght.

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With compass, simetral of the lenght with semicircles.

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One half of the lenght in range of the compass. Semicircle.

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Where lenght’s simetral intersect semicircle is the center of described lenght’s circle.

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Lines from the ends of lenght, and through the center of descriptive circle of the lenght „intersect“ circle’s arc and divides it on 4 parts.

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In that way, quadriangle is made, correctly and simply.

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As the lenght’s simetral intersects circle’s arc, we take intersectiones of lines that form quadriangle in the range of compass.

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That range (radius) divides circle on 8 parts. That is one of the universal principles of construction of quadriangle and octogon.

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We are interested in semicircle of given lenght whoch we used at the beginning for construction of quadriangle. Its radius matches quadriangle’s descriptive circle. Take a look what it does on the quadriangle’s descriptive circle.

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Start from simetral’s peak all the way till it retur nin it by division (tour). Divides descriptive circle of quadriangle on 26 parts or 2 x 13. Therefore, now you can do with number 13 all its sequence (3 times, 4 times, 5 times, 6 times, etc.).

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Other way : lenght. Lenght described with semicircles.

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Equitentacle triangle, sides of lenght’s size. Triangle’s simetrals. Simetrals intersect simetral’s semicircles. There is the center of basic’s descriptive circle (lenght).

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We describe circle from that center. Where simetral’s semicircles intersect circle are quadriangle’s peaks.

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As the lines divided circle on 2 x 4 or 8 parts, inscribe semicircles of every third from every pole.

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We’ll analyze just one. What does that radius on descriptive 4 and 8 ?

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Divides it on 9 parts. Now a little independance practice. If the peaks of equitentacle triangles are the center of base’s descriptive circle, on how many parts base’s lenght divides those circles (lined) – it would be a little repetition of past chapters.

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An afterword of the first part

Many of you must ask yourself why you don’t learn this many ways of construction of polygons in schools because you concluded it makes sense. Simple. They didn’t know. And in concept of protractor, they thought it shouldn’t be. Second. In mathematics greater concetracion is on number, and geometry is the same thing. Number doesn’t form visual picture. For sure, it is equally worth because it confirms or doesn’t confirm, but visual picture. Through eye. Further is process about which we don’t know much, yet. But, someone asks how I know it. I don’t know. Nobody tought me. It would be strange to someone when I say I am also a kind of pupil. I must admitt. Something or someone invisible „pushes“ me in idea, and I must solve untill I understand it visually. I bring it step by step with drawings to you. You have to know one thing. Before and not so long time ago, special skills were worth, so through ages many have hide it and save for their groups, so called secret societies. It is truth that in ancient times special people and groups were responsible for that knowledges whose mission was to use that knowledge for benefit and development of nations. But trough ages it turned out wrong. Many posseses knowledge today, but only for their benefit (unfortunately). Another thing. Your teachers can’t learn you different, because they have regulations. And those who make regulations are, unfortunately, lazy or ignorant. How to understand fact that nothing changed from Greeks till today on geometric field. 2500 years. Unbelieveble. But let people to discuss with his own consciousness and conscience. Take a look at the concept of octogon if we have base of some size.

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Line. Given lenght on it. Describe it with semicircles. Equitentacle triangle and its simetral, and from its peak we describe circle of given lenght radius – base.

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Simetral divides circle on two parts and take that intersection as peak pole of circle’s division with its radius on 6 parts.

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Inscribe its star hexagonal polygon.

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And then to the star hexagon its smaller star polygon.

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So, we found circle’s center which, in the beginning, will be divided on 8 parts with given base’s lenght.

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So, from that peak of equitentacle triangle we describe given lenght by circle – base. In range of compass we take base, given lenght and from its final points we start division of circle’s arc.

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So. We divided circle on 8 parts. Center of descriptive circle is correct.

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An afterword

It is (next chapter) when one doors, on which you „knocked“, open. It means a question : how to find center of some regular polygon if we have some base’s lenght, just with compass and ruler without measures. Soon you see what is and where is code system for all regular polygons. In that way disappears unknown quantities of geometric nature. Then you understand what our wise people said. You learn all your life. There is always something new. We use to add that in the end you don’t know anything. It is not truth. What you learned makes further base for new. Our mistake is we want to understand final product if the end exists or find the last answer. It is in our human nature. We are „running forward“trying to theorise that final, and we are in danger to „entangle“ in esotery or unproof. Therefore, step by step, not easy school when life is teacher. I tell you from personal experience. They ask me and sometimes I ask myself, what have you been doing, what did you do. I would say a lot and nothing, but say for yourself – is all this for nothing?

HR – RIJEKA 12.01.2013.

One Response to “Quadriangle and octogon – Childrens’s education”

  1. Ricardo says:

    The Seed of Life = 1 x 2 x 3 = 6 … – my e-mail if anyone wants to chat about it.
    The Flower of Life im not sure but i believe its 3 x 6 x 9 = 108, triple of the seed i believe. Its getting interesting Mr. Tomo, very interesting.

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