Doubling cube
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Doubling cube
With de properties of the circumference

Preamble: about constructive numbers (Pi and cube root of 2)

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Since ancient times there has been an attempt to square the circle and double the cube among other problems, reaching the conclusion that it is impossible to do so for many years.
Among many mathematicians who support, develop and explain this conclusion we could mention L. Euler, J. Lindeman, Weierstrass, etc. and almost all the philosophers and mathematicians of antiquity including Greece.
Their final deduction would be:
"Both the number Pi and the cube root of 2 are non-constructive numbers from a unit segment, (ie. r = 1)"
And this can be developed by means of deductions and explanations concerning the properties of irrational numbers, etc.
Well, so far everything is perfect, except for one question: They have not taken the correct path to square the circle and doubling the cube:
"The segment Pi, and the segment cube root of 2 are already constructed and inserted or inscribed within the circumference, and therefore: It is not necessary to build them"
We just need to search for them, find them and draw or trace said segments.
And how it is done?
Very easy: finding its position and limits through the appropriate arcs and segments.
And then, support us on these well-known arcs and segments, check that their exact value coincides with the one already known in mathematics.
Therefore, in order to begin to understand the problem better, we consider the previous statement: Pi and the cube root of 2, are two segments that are already defined and inscribed within the circumference, and we only need to find them.

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Fig. 1

In this study we are going to try (and achieve) to obtain the duplication of the cube, a traditional problem posed since ancient Greece, and which was always considered Unattainable due to the impossibility of obtaining a segment identical to the cube root of 2 (1.259921. .....), necessary to build a double cube of another dice.
However, my thinking and philosophy has always been that we should not put borders on mathematics, and therefore not on the impossibility of providing solutions to these three classic problems, such as the trisection of angles, squaring the circle, or this one of cube duplication.
Therefore, trying to find a solution, I have been checking the circumference to see if it could provide us with the solution.
And yes, surprisingly, within the circumference (r = 1) there is that segment that is the cube root of 2, or if we want, the cube root of its diameter.

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Fig. 2

For that, we are going to show where the segment we are looking for is located (C = cube root of 2). For this we see the drawing (fig. 2) in which, and within the circumference, we can consider the following elements:

1.- In the first place the radius = 1 of the circumference and its diameter 2, whose cube root we are looking for.
2.- Second, and starting from the beginning (A) of the diameter of the circumference, we draw a chord of circumference C (Segment AB, which in the practice of the drawing, and to facilitate the work, will be replaced by the rule).
3.- Immediately we see that this chord C has its projection on the diameter in the segment (p), which forms the third element necessary to find the cube root of 2 (C).

Well, if we observe and work with two of these three elements, chord (C) and its segment of projection (p), we construct a formula that is basic and essential for our investigation:
C.p = (from 0 to 4).
But attention, when the formula gives us as result the radius of the circumference C.p = 1 (third element), then we measure C and see that it is exactly the cube root of 2, that is, the segment we were looking for. (1.259921 ... x 0.79370052 .. = 1).
Therefore, we already observe that there is an interrelation or quadrature through segments and formulas between the radius (or diameter) of the circumference, with the cube root of 2, and its projection on the diameter.
At this point we can already begin to feel some concern and intrigue:
We will be able to obtain or "hunt" the elusive segment C, that is, cube root of 2.?
Well, we do not worry about, the answer is yes, even fun to do.

But how logic would say that we have to do it?

Well, using the component elements of the previous formula, C.p = 1, but using a rule and compass instead of numbers and quantities.
- The chord-segment will be replaced by the rule with which it is supported at the beginning of the diameter (A) until the circumference is cut at point (B).
- The projection segment on the diameter (p) will be used as arc (arc p)
- The radius (1) will be used to create another connecting and measurement arc (arc. r) that will be the unit of measure and future side of the cube to be duplicated.

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Fig. 3

Then (fig. 3), all these basic elements of both the formula and geometric drawing elements will bear fruit and results when the three converge between them at point (q).
When this happens, point B is marked and also segment C = AB, which will have acquired the dimension and value of the cube root of 2.

Therefore, the resolution of the problem is to hunt down the point (q) of confluence of these three elements.
For that reason, and in my opinion, it can be considered fun, because the main and only job will consist of the pursuit, capture and hunting of the element or point(q).

Practical work:

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Fig. 4

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Fig. 5

In this section, we should already forgotten a bit about numerical measurements and checks: Let's take the compass and the ruler, and proceed to draw on a circumference of radius 1, with its center and diameters already marked.
- The first thing to build is an arc with radius = 1, (with center in point A) and from de center of the circumference to the left, to serve as a guide, measure and support for the other two basic elements that we are using.
- Now, we take the ruler and on the beginning of the diameter (point A), we cut with it the circumference at the test point (B), creating a space or segment from this point to the vertical diameter.
- At this segment or distance (a-a'), we measure it with the compass, and we translate this measurement over the horizontal diameter creating the segment (Oh = a-a'), at which point (O) we begin the verification of the quadrature and confluence of elements with the compass.
Logically the first attempt will fail and there will be no coincidence between the rule and the cut between the arches, (cut arc r - arc p).
So we move the ruler in the right direction and try the squaring or confluence of elements again.
Moving the ruler in the right direction and making the previous checks, after 3-5 attempts, the quadrature will have been achieved and we will be on the point (q) of confluence of these three elements.
At that moment, we mark point B on the circumference and we can definitively draw the segment AB = C which is the cube root of 2.
With this result, we can now build our reference cube, whose side will be the radius of the circumference r = 1, and apart, the duplicated cube with side 1.259921 ... which will be twice as large as the previous one.

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Fig. 1

On the cube roots of segments

This is a job that I have not done, however and in principle, I think that with this method we can already do any kind of roots on segments, always taking into account the aspect ratio, since here we have used the unit radius r = 1, and for other measures of the segments, these will have to be adapted by the ratio between units used, as well as the power of this ratio.

Method of matching and confluence of arcs and segments

"In geometric figures, equality, duplication and coincidence of various segments and arcs can occur, which can help us to achieve the correct construction and measurement of said figures."
As we have seen, this method can achieve cube duplication and angle trisection.( ferman )