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Secondary Magazine - Issue 47: Focus on

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 12 November 2009 by ncetm_administrator
Updated on 26 November 2009 by ncetm_administrator

Secondary Magazine Issue 47compass, pencil, ruler

Focus on...Constructions

There are three construction problems using only a straight edge and a pair of compasses that the ancient Greeks failed to solve, and that have since been proven to be impossible:
  • trisecting the angle
  • doubling the cube
  • squaring the circle.

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In 1796, a 19-year-old student at Göttingen University, Germany, proved that it was possible to construct a regular heptadecagon (17-sided polygon). This discovery may be the thing that tempted him away from his planned study of languages to mathematics, eventually becoming one of the greatest mathematicians of all time. His name? Carl Friedrich Gauss.

heptadecagon construction - animation by Jonathan48 

Animation by Jonathan48. 

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Gauss went on to show that a regular polygon with n sides is constructible if n is the product of a power of two and one or more Fermat Primes. The proof that these were the only constructible regular polygons was given by the French mathematician Pierre Wantzel in 1837.

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Although Gauss proved that it was possible to construct a regular 17-sided polygon, there’s no evidence that he actually went on to do it! The first recorded construction of a heptadecagon is by Johannes Erchinger a few years after Gauss’ proof.

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Gauss concludes his 1801 work Disquisitiones Arithmeticae with the list of numbers:

3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68,
80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272.

This is the list of the number of sides (under 300) of regular polygons that are constructible.  

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Constructions for a regular pentagon were described by Euclid in his Elements around 300 BC. He also described how to use this construction, with one for an equilateral triangle, to construct a regular 15-sided polygon.

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Squaring a circle, that is, constructing a square with exactly the same area as a given circle using only compass and straight edge, was proved to be impossible in 1882 since it requires the construction of a square of side \sqrt{\pi } . This proof, finally laying to rest one of the three construction problems of antiquity, was a sub-result of the proof by the German mathematician Carl Louis Ferdinand von Lindemann that \pi is a transcendental number.

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The (impossible) problem of doubling the cube, that is, creating a cube with a volume twice that of the original, is also known as the Delian Problem. According to Wikipedia,
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 some sources however it was the citizens of Athens who consulted the oracle at Delos.) 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. In any case the story is almost certainly fictional, at least in most of the details. According to one theory, the ancient Hindus had devised similar problems involving altars, and this version spread to Greece.

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