Cube Root of 2. Double The Volume of A Cube. The Delian Problem

Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first.
In algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 = 2; in other words, the cube root of two, means, what number, multiplied by itself 3 times, will give the answer of 2. This is because a cube of side length 1 has a volume of 13 = 1, and a cube of twice that volume (a volume of 2) has a side length of the cube root of 2 which is 1.259... That is, 1.259... x 1.259... x 1.259... = 2

This discussion shows up in Plato's Republic (c. 380 BC) VII.530

History
The problem owes its name to a story concerning the citizens of Delos, who consulted the oracle at Delphi in order to learn how to defeat a plague sent by Apollo. According to Plutarch it was the citizens of Delos who consulted the Oracle at Delphi, seeking a solution for their internal political problems at the time, which had intensified relationships among the citizens. The oracle responded that they must double the size of the altar to Apollo, which was a regular cube. The answer seemed strange to the Delians and they consulted Plato, who was able to interpret the oracle as the mathematical problem of doubling the volume of a given cube, thus explaining the oracle as the advice of Apollo for the citizens of Delos to occupy themselves with the study of geometry and mathematics in order to calm down their passions.

Origami may also be used to construct the cube root of two by folding paper.

In music theory, a natural analogue of doubling is the octave (a musical interval caused by doubling the frequency of a tone), and a natural analogue of a cube is dividing the octave into three parts, each the same interval. In this sense, the problem of doubling the cube is solved by the major third in equal temperament. This is a musical interval that is exactly one third of an octave. It multiplies the frequency of a tone by 24⁄12 = 21⁄3 = 3√2, the side length of the Delian cube.

Jain 108

(From Wikipedia, the free encyclopedia)

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