The Problem of Doubling the Cube

Abstract:

According to legend, sometime in the fourth century BC the people in the Greek city of Delos were suffering from a terrible plague. Relief was sought from the oracle at Delphi.

Delos had an altar in the shape of a cube that was used in religious ceremonies, and the oracle told the people of Delos that they could rid themselves of the plague if they constructed an altar double the volume of that altar.

How would one do this? Some the most brilliant of the ancient Greek mathematicians worked on this problem. Did they solve it? We present two perspectives.

Part 1 (Karl Zimmermann): The Greeks were interested in constructing geometric figures using only a straightedge and a compass. However with these tools, they were unable to double the cube. We'll take a modern approach to this problem to show that it is, in fact, impossible.

Part 2 (Julius Barbanel): We present two methods of doubling the cube that involve techniques beyond the straightedge and compass constructions of Part 1. One is due to Eratosthenes (276-194BC), and is a sort of mechanical argument, involving moving triangles. The other is due to Archytas (428-350BC) and involves a three-dimensional construction in which a certain point is determined as the intersection of three surfaces of revolution.

Union College Math Department Home Page Comments to: math@union.edu Created automatically on: Mon Jul 23 03:37:59 EDT 2018