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Cube (the minifying enigma) – Practical Application


In all this turbulence, especially on occasion of analyzing the cube (a common geometrical body with a strange history), it crossed my mind that in the case of concrete material matters it is always necessary to also act concretely (the way that matter “is resolved” for me and indeed for everybody else who have been following my last chapters. Hence, it is of essence to concretely put practical application into action, starting from the youngest “geometricians” in the elementary and secondary schools. Why start the practical applications with them? The answer is simple. How can anybody expect academically educated mathematician to do the work of the “masonic profession”? It would be considered below his level (don’t lay the blame on me, but like it or not, that is our present-day reality). But, it doesn’t matter. Therefore I have decided to create a small program that will plainly “tell” how all this can be put into action, I dare say, in a somewhat less complicated way; by minifying the cube only with the compass and unmarked straightedge although the concretization of duplication is just as simple, but the process of diminution is terminal, whereas duplication is interminable. Or…, but anyway, be it as it may, you pupils and everyone else should follow these instructions to see that this is so. Anyway, you can check this. So let’s start with the realization of the instructions. It’s best to use a cube of solid wood or plastic. They are most easily processed if in one piece and the compass is the most effective and accurate instrument to use.

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To repeat: the cube is a hexahedron with six equal squares as faces. How do we reduce it to be twice smaller (once) only with the use of the compass and unmarked straightedge, in other words in compliance with the formula r divided by 2 (singly). This that follows on one square surface should be drawn on all six faces, only with a compass and unmarked straightedge (universal code – hence you can use a cube of any random size).

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Thus: (I reiterate, on all 6 sides of the cube) we delineate with diagonals in order to find the midpoint of the quadratic surface.

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With diagonals we find the midpoint using the compass, then we draw the inscribed circle of the cube’s quadratic side (reiteration: what we are now showing should be depicted on all six sides of the cube, best immediately).

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Note: Even the inscribed circle of the cube’s square doesn’t matter, because the important point is the division into 24 parts all the way to its edges. Therefore, with the same radius in the compass, and divided intersections by diagonals and the circular arc, we divide the circle into 12 parts.

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And since this division is not ideal for further divisions into 24 parts, we will inscribe four equilateral triangles or a twelve-sided rectilinear polygon circle.

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Thereby we have created conditions to diminish the cube once, twice, thrice and so on. (Once again, I reiterate that the size of the inscribed circle does not matter but rather its division into 24 parts of extended directions up to the edges of the cube, but note that this should be depicted on all 6 sides of the cube.)

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Now, when we have the divisional groundwork to the edges of the circle into 24 parts, we connect the first angles with line segments and they create intersections with diagonals that give us r:2 or a unilateral diminution of the cube’s side or a one-sided decrease of the cube.

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We will extend the reduced cube to the edges (unilaterally) and thus attain points of the cube that we can “cut” to get the diminished cube. In order to give you a clearer picture we will show a couple of reductions for the purpose of underlining the principle.

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Duplicated reduction of first cube and its unitary reduction. Unitary reduction of an already unilaterally reduced cube. We will let all the reductions of previous reductions remain because they will be needed for further decreases and explication of the principle of further reductions.

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The third reduction of the first (cube) and the second (cube) reduction (in a somewhat bolder presentation for the purpose of easier detection).

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In the case of the fourth reduction it can also be presented in this way (by a dotted line), or simply by showing the intersections of the 24-sided polygon division of the previous drawing.

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The fifth reduction. And this goes on and on and ever coming closer to its termination; to the seventy-seventh (77th) reduction which is the null point, respectively.

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Certainly we should not have gone further into more than one reduction, but in order to make the principle more understandable it will come in handy. How? Take a calculator into your hands, measure the length of the sides of the cube and constantly divide them by 1.259921, which is the cube root of two. And how the side of the cube is actually the radius of the inscribed hexagon of the circle, the formula for reduction is r (radius) divided by 2 (with the cube root of two). Note down each following result until you get to the null point. In this way you will discover when the transition into less than 1.259921 takes place, but just go further on. However, this might be a tedious job and it also depends on the numerical size of the radius but is relatively simple, indeed. For example: if the size is 1, then it is 66+1; and when it is 2 = 66+2, etc. However, this is for research of those interested in exploring numerology and its regularities, and our objective is the geometrical path only with an unmarked straightedge and compass, the ancient regularity of natural concordance.

I hope to soon be able (indeed, due to a financial shortage, I don’t know when) to show you the duplication of the cube in a similar way.

Croatia – Rijeka, August 14, 2014
Author : T. Perisha
Web; Slim
English: S.F. Drenovac

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