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Doubling the Cube (education for elementary school kids)

CHAPTER 15

It’s a good thing that complete simplicity exists, even though it’s the hardest thing to find, especially when a lot of water has flown under the bridge since we’ve stopped being kids and are probably much closer to coming to the end of the road, no longer playful, too serious and overfilled with impressions and pressures of all these worldly crises: we, who are forever looking for answers of why and what is the reason and how come it is so and not otherwise? Therefore kids, don’t hold it against me if I often forget that we also used to kids in our own timeline, a time that was much more leisure than it is today, but that’s how it has always been the case throughout all the past centuries for the generations that preceded us. In all likelihood they thought the same as I do now, imagining myself as their hope in pursuit of some new world, a world of simplicity and the greatest possible ease. For ever and ever there has always been some kind of an easier and simpler way, but remember that such a way is the hardest to discover. Hence, to shorten this somewhat complicated foreword, otherwise you might wonder “what’s this elderly guy ‘babbling’ about”, and thereby be doing wrong to my “gray-headed self”. Respect your elders for you too will one day be sharing the same destiny. But now let me reveal to you the discovered simplicity of an ancient, they say ‘enigma’, in the simplest and most unproblematic elementary-school geometry way, only with the use of a compass and unmarked straightedge, step by step.

* * *

41501

Thus the circle of any radius entered into the compass, divided by that radius, separates it into 6 parts. This is a geometrical regularity.

* * *

41502

If we connect every other pole with line segments we will acquire a star that some call „The Star of David“, actually a hexagonal star-shaped polygon.

* * *

41503

Since the line segments intersect, the opposite intersections enable the division of the circle into six more parts, which means into a total of 12 parts.

* * *

41504

And if we join the opposite intersections again (to avoid complicating the matter, we draw only one), inside the circle we then get a quadrilateral (square) with sides the magnitude of the radius that we entered into the compass to draw the circle.

* * *

41505

The quadrilateral aids us to further divide the circle, but for the moment the division of the circle is not of essence for the enigma, but the inscribed hexagon polygon. Therefore, we join the first 6 poles of the circle with line segments.

* * *

41506

With straight lines we connect the opposite poles and with line segments reinforce (three of them – two near the top and the opposite ones next to the center). Then we make a cube of sides (edges) the length of the radius, and the square in the center is one of its sides.

* * *

41507

Lines as diagonals intersect the cube. We join them with line segments so as to get a square that is larger than the first one. It is doubled, a doubled cube equal on all sides. Now we need to verify this, and this will be your homework, while we will move on to assignments for secondary schools.

* * *

41508

Although you get instruction on inverse exponentiation in the last elementary school year, and if you have learned it, the formula for this doubling is: radius times cubic root of 2 or r 1.259921. Now we enter the new radius into the compass and from the same midpoint we will draw the circle of that radius.

* * *

41509


Hence, the circle is the radius:
The first radius times cubic root of two.
Its square, therefore the cube’s edges are equivalent (dotted lines).

* * *

41510

The cube and its magnitude likewise, but from a front view aspect, thus as a top view and side view; but is that the case as a three-dimensional object (equal on all sides)?

* * *

41511

The derivative of three-dimensional objects is, no doubt, simple from a front view aspect (dotted lines), a doubled object of one within the other and equal from all sides, respectively.

* * *

41511a

But from a perceptive aspect it is relatively hard to grasp on a two-dimensional surface, thus models are used on addition to the front view projection, something that will already as of now be simplified by simulation via computer animation. Therefore we will rather focus attention on the principle of duplication.

* * *

41512

It is the principle of the first duplication by way of a circle and the division of its sides (into 24 parts) as radiuses.

* * *

41513

And that can carry on invariably. This (universal) principle can perform duplications invariably.

* * *

41514

Diminutions arise from the third duplication by way of a siple principle (dotted line).

* * *

41515

Constantly, until it makes a transition into negation (via a system of division: radius divided by cubic root – the third of number two).

* * * *

CONCLUSION

Therefore we may conclude that this will suffice for the time being. The principle of doubling the cube and its diminution is likewise simple from the standpoint of geometry only with a compass and unmarked straightedge. The purpose and meaning of “unknown” becomes a more or less philosophical term (Platon). Its chemical aspect suggests expanding and shrinking or an aggregated condition. But there is no need to cogitate on purpose and meaning. For the time being it will do to say “it is simply and naturally so”. So I won’t bother you with my thoughts about all of this and instead I’ll simply conclude: “Once upon a time there used to an enigma. But it is no more.”
Attention: regardless of a twin similarity with the circumscribed circle, there is a minimal decimal divergence and it is good that it does not identify, not even geometrically. In accord with a new custom of mine, I won’t “make a present” for you with a few of my 350 thousand verses, but instead I recalled a few verses by poet Branko Radichevich from my own school-days who described that period concisely, simply and clearly:

“From the cradle and until we die
school-days are the best time of our life”

Croatia – Rijeka, July 11, 2014
Author: T. Periša
Web : Slim
English: S.F. Drenovac

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