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Geometric solid(s) – The cube and its division

CHAPTER „C“ FOR KIDS

Geometric solid(s) – The cube and its division

In this Chapter we will get acquainted with the construction of the CUBE in the ancient way with a compass and an unmarked straightedge. This solid is the element that is, or should be, the fundament of human activity. In the beginning we will divide it into its simplest parts, and thereafter in the forthcoming chapters into all of its other parts. Why is it the fundament of human activity? Because the proper construction of the cube will lead us to a series of “unknowns” that have been disregarded and which would bring us to the recognition of the natural harmony between man and his activity in nature. Yet so far we have not made much of an effort to explore this relationship and teach it to ourselves and our descendants, but on the contrary, exultant with our technological orientation, natural harmony was omitted, thus creating a chaos of unnatural figures and forms of whatever comes to mind, all the way to the crazy conviction that with technology i.e. techno-gadgets we can do anything we want.  How erroneous such a conviction was best felt by a Japanese company who wanted to erect a replica of the pyramid by techno-technological appliances, and the whole venture was a flop.       On the other hand the cube is a code system without which ancient enigmas cannot be solved, such as doublings of squares and cubes, the relation of measurements because they solve the progressions of halves, thirds, fifths, sevenths by geometrical and congruent progressions. We should mention that ancient peoples had no knowledge of decimals. And what is a decimal? A fraction. And a fraction? A part of a whole. So, a part can be drawn, without words, by the ancient means of division. Division of what? The cube, and this can be the diametrical square or cubical – only with a compass and an unmarked straightedge. Therefore we won’t waste much of our time on present-day “discursive” constructions of the cube, but will deak with the way that ancient geometry teaches us.

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Although they have the same length of edges (even this isn’t taught today), we get some arbitrary parallels by which a shape of three dimensions is produced to create the impression of the geometrical solid of a cube. Then isn’t it a pity to waste our time on this?

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The cube – in the ancient manner (compass & unmarked straightedge) as shown in chapter „B“ on the construction of the square. Therefore, the given length is the radius. The same goes for the cube. Describe a circle of arbitrary length of the cube’s edges, and divide it with circles of the same radius into six parts.

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- Connect poles with lengths. We acquire the circles of a hexagon.
– Connect the subtended poles. Cube (the eighth edge of the cube is hidden behind the front edge). Now we divide it.

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The centerlines of the sides are already inscribed when we partition the circle with circles of same radius. Those are their exterior intersections and among other things they serve as control points for correct drawing. Therefore, subtended intersections are connected by straight lines (directions). ( These directions pass through the center).

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The intersections of the straight lines and the cube’s edges are the points that we connect in the direction of the edges. We divided the cube on the cube of 3, i.e. 23 – 2 x 2 x 2. This is one of the ways.  Now we can go further on with the progressions of 2.

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Another way, albeit shortened, is used only when an enlargement of the presentation is desired. Thus, the inscribed hexagonal circle of radius of the arbitrary value of the cube’s edges – cube, star-shaped hexagonal polygon (every other pole). Thereby we create the conditions to divide the cube 2 x 2 x 2 = 23 .. Its further division is logical.

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But with the hexagonal star-shaped polygon we also created conditions to divide the cube on 3 x 3 x 3 = 33 and onward. But we will stop here and show the enlarged (shortened) form. Hence, circle of radius length of arbitrary edges – hexagon – cube – star-shaped hexagonal polygon.

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Now we connect the intersections of the star-shaped polygon – first left and right from the diameter (parallel with the diameter) – vertically.

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- The other two intersections are parallel with the other diameter since the hexagon has three diameters. Those lengths should go to the edges of the cube…

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- … and the third two parallel with the third diameter of the hexagon…

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… and now we just connect the acquired points on the edges. We divided the cube of 3 as follows: 3 x 3 x 3 = 33 and its radius (radius of the circle) into 3 parts, and the diameter into 6 parts (thirds and sixths). That would be enough for now, since we still need two important basics (the fifths and sevenths).

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