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Doubling the cube (also known as the Delian problem) is one of the three most famous geometric problems unsolvable by compass and straightedge construction. It was known to the Egyptians, Greeks, and Indians.[1]

To "double the cube" means to be given a cube of some side length s and volume V, and to construct a new cube, larger than the first, with volume 2V and therefore side length s\cdot\sqrt[3]{2}. The problem is known to be impossible to solve with only compass and straightedge, because \sqrt[3]{2} is not a constructible number.

Contents

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.[2][3] (According to some sources however it was the citizens of Athens who consulted the oracle at Delos.[4]) The oracle responded that they must double the size of the altar to Apollo, which was in the shape of a cube. The Delians consulted Plato who in turn gave the problem to Archytas, Eudoxus and Menaechmus who solved the problem using mechanical means; this earned a rebuke from Plato for not solving the problem using pure geometry. However another version of the story says that all three found solutions but they were too abstract to be of practical value.[3] In any case the story is almost certainly fictional, at least in most of the details. According to one theory,[4] the ancient Hindus had devised similar problems involving altars and this version spread to Greece.

A significant development in finding a solution to the problem was the discovery by Hippocrates that it is equivalent to finding two mean proportionals between a line segment and another with twice the length.[3] In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that

a:r=r:s=s:2a.\

The first mathematician to rigorously show that the problem is unsolvable by compass and straightedge construction, was Pierre Wantzel (1837).

Solutions

An illustration of the ruler-and-compass method.

There are many ways to construct \sqrt[3]{2} which involve tools other than compass and straightedge. In fact, some of these tools can themselves be constructed using compass and straightedge, but must be cut out of a sheet of paper before they can be used. For example, following Sir Isaac Newton, construct a ruler with a single unit distance marked on it. Construct an equilateral triangle ABC with side length 1, and extend side \overline{AB} by one unit to form the line segment \overline{ABD}. Extend side \overline{BC} to form the ray \overrightarrow{BCE}, and draw the ray \overrightarrow{DCF}. Now take the ruler and place it so that it passes through vertex A and intersects \overline{DCF} at G and \overline{BCE} at H, such that the distance GH is exactly 1. The distance AG will then be precisely \sqrt[3]{2}.

Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the cube involve the cissoid of Diocles, the conchoid of Nicomedes, or the Philo line. Archytas solved the problem in the fourth century B.C. using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution.

False claims of doubling the cube with compass and straightedge abound in mathematical crank literature (Pseudomathematics).

References

  1. ^ Lucye Guilbeau (1930). "The History of the Solution of the Cubic Equation", Mathematics News Letter 5 (4), pp. 8–12.
  2. ^ L. Zhmud The origin of the history of science in classical antiquity, p.84, quoting Plutarch and Theon of Smyrna
  3. ^ a b c T.L. Heath A history of Greek mathematics, Vol. 1
  4. ^ a b D.E. Smith, History of Mathematics Dover (1958) p. 298

External links

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