Geometric Problems of Classical Greece: Dice Duplication – 01/26/2021 – Marcelo Viana

Because of the insights from Mesopotamia and Egypt, ancient Greece made remarkable advances in geometry. However, three problems withstood the ingenuity of the Greeks and possessed generations for more than two millennia: doubling the cube, dividing the angle into three, and squaring the circle.

The three were not understood until the 19th century, when it could be shown that they are all insoluble. This did not prevent curious people from continuing to produce “solutions” (I get several a year to give my opinion).

The problem with duplicating the cube is to create the edge of a cube that is twice the volume of a particular cube, using only a ruler (not graduated) and a compass.

According to the historian Plutarco, the problem originated in a consultation between the island of Delos and the famous Oracle of Delphi on how to appease the god Apollo who had plagued the island. The oracle replied that they should duplicate the altar in the Temple of Apollo, which was a cube. They immediately replaced the altar with another double-edged die, but the plague continued. When consulted, the philosopher Plato clarified that it was the volume that had to be doubled, not the margin, and interpreted that Apollo recommended the citizens of Delos to spend more time on mathematics. That seems to me to be excellent advice from God, by the way.

According to Plutarco, however, Plato would have passed the problem on to the mathematicians Eudoxus, Archytas and Meaechmus, who used various mechanical instruments to find solutions that made the philosopher catch the tune of the three because they didn’t just use pure geometry. that is, ruler and compass.

From a mathematical point of view, the problem is to construct a line segment whose length is a cube root that is twice the length of a given segment. The solution was given in 1837 by the French Pierre Wantzel (1814–1848), who observed that a segment can only be constructed with a ruler and compass if the length is the root of a finite chain of equations of degrees 1 or 2 Cube root of 2 so the problem of doubling the cube is impossible.

In the same work Wantzel also proved the impossibility of a tripartite division of the angle. Six years later, he was still showing that if a third degree equation has three real roots, its solution necessarily involves imaginary numbers. This curious fact went a long way towards establishing the legitimacy of imaginary numbers.

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