I had the problem, that I do not see how to construct the angle at the base of the pyramide. However I thought, that things become easier if one simplifies the geometry.
Instead of the pyramide, one might take a prisma made of two orthogonal triangles S1-B-A and S2-C-D, with right angles at A and D. Again AB = BC = CD = DA and S1A= S2D = h .
Then V = 1/2 h L**2 with h = k L, gives V = 1/2 k L**3 = 2 a**3 => 1/2 k = 1 => h = 2L, which fixes the angle at S1 and S2.
The apparent contradiction, that there are solutions to unsolvable problems, seems to be due to the fact, that algebraic theory requires more precision, then what you can do by trying to fit something together in a practical way. As far as I can understand, it is proven, that by employing ruler and compass, you cannot construct cubic roots (basically because the equation of the circle is only quadratic). So the question remaining is: Have you invented another "dirty" method to solve an unsolvable problem, or have you postulated "volume-pooring" as a new geometrical construction method, which adds cubic roots to the class of constructible irrational numbers?
(I am no mathematician. It just happens, that I am reading currently a book touching these questions.)
1. This reversed pyramide without bottom was invented for finding cubic roots (filling its volume by liquid - which volume is equal volume of two cubes - value that we seek - is the side of the square on the upper surface of the liquid in the pyramide). I explain how to use the pyramide for solving the problem of duplicating of the cube. But i don't understand, how can be used for this problem a prisma you suggest (you don't described any method of finding a cubic root). 2. "...because the equation of the circle is only quadratic..." - it's true for plane, but volume of the pyramide in three dimensions is cubic, and it is significantly used in this method. For example, with this pyramide you can't find any root of fourth or fifth degree or upper. 3. I prefer to call this method "direct measurement of the volume and splitting along one coordinate". The pyramide was well known to Greeks, and measuring the volume with displaced fluid was used by Archimedes (remember his famous law) - new here is only combination of these method for solving an old unsolvable problem. 4. I can't decide, is this method dirty or not. Using this method for calculating cubic roots, imho - not practical. For today the only use of this method i can imagine - saying several words about it on geometry lecture for university students.
>(you don't describe any method of finding a cubic root).
Of course, I do not. I am basically doing nothing else but using your method. Rewriting it for myself has clarified the things for me. I am now convinced that one can either use an inverted pyramide or the prisma (put on its tip) in rather equivalent ways.
I am basically using your equations. (I am not sure whether my way to write potencies is generally understood: a**3 = a * a * a).
The only other difference is, that I replace h in the volume formula in a different way. Thereby you calculate k = SA/L, while I calculate something I should not have called k, but may be q = h/L.
For the pyramide I get h = 3L, which is consistent with k = (9 + 1/2)**1/2.
Instead of the pyramide, one might take a prisma made of two orthogonal triangles S1-B-A and S2-C-D, with right angles at A and D.
Again AB = BC = CD = DA and S1A= S2D = h .
Then V = 1/2 h L**2
with h = k L, gives
V = 1/2 k L**3 = 2 a**3 => 1/2 k = 1 => h = 2L, which fixes the angle at S1 and S2.
The apparent contradiction, that there are solutions to unsolvable problems, seems to be due to the fact, that algebraic theory requires more precision, then what you can do by trying to fit something together in a practical way. As far as I can understand, it is proven, that by employing ruler and compass, you cannot construct cubic roots (basically because the equation of the circle is only quadratic). So the question remaining is: Have you invented another "dirty" method to solve an unsolvable problem, or have you postulated "volume-pooring" as a new geometrical construction method, which adds cubic roots to the class of constructible irrational numbers?
(I am no mathematician. It just happens, that I am reading currently a book touching these questions.)
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I explain how to use the pyramide for solving the problem of duplicating of the cube.
But i don't understand, how can be used for this problem a prisma you suggest (you don't described any method of finding a cubic root).
2. "...because the equation of the circle is only quadratic..." - it's true for plane, but volume of the pyramide in three dimensions is cubic, and it is significantly used in this method. For example, with this pyramide you can't find any root of fourth or fifth degree or upper.
3. I prefer to call this method "direct measurement of the volume and splitting along one coordinate".
The pyramide was well known to Greeks, and measuring the volume with displaced fluid was used by Archimedes (remember his famous law) - new here is only combination of these method for solving an old unsolvable problem.
4. I can't decide, is this method dirty or not. Using this method for calculating cubic roots, imho - not practical.
For today the only use of this method i can imagine - saying several words about it on geometry lecture for university students.
Reply
Of course, I do not. I am basically doing nothing else but using your method. Rewriting it for myself has clarified the things for me. I am now convinced that one can either use an inverted pyramide or the prisma (put on its tip) in rather equivalent ways.
I am basically using your equations. (I am not sure whether my way to write potencies is generally understood:
a**3 = a * a * a).
The only other difference is, that I replace h in the volume formula in a different way. Thereby you calculate k = SA/L, while I calculate something I should not have called k, but may be q = h/L.
For the pyramide I get h = 3L, which is consistent with k = (9 + 1/2)**1/2.
Reply
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