Constructible Angles and Regular Polygons
Date: 04/17/98 at 22:20:26 From: Danny Dawson Subject: Euclidean Construction of Polygons I've found that you can create angles (using Euclidean Geometry) that are multiples of 9 and/or 15 degrees. I've used the construction of regular polygons to help me, but that's all I can construct. Are there any other angles that are constructible, and what regular polygons are constructible? I'd appreciate a prompt respose because I've been collaborating with a reporter on an article for National Mathematics week. Thank you.
Date: 04/20/98 at 11:56:10 From: Doctor Wilkinson Subject: Re: Euclidean Construction of Polygons It was very clever of you to notice the connection between construction of regular polygons and construction of angles. The problem of which regular polygons can be constructed was solved completely by Gauss around 1800, although there are still some mysteries connected with it. The angles you mention correspond to the regular hexagon and the regular pentagon. But there are other constructible regular polygons, notably the ones with 15 and 17 sides. The rule Gauss discovered was that a regular polygon with n sides can be constructed if and only if n is the product of a power of 2 and of distinct primes of the form 2^2^k + 1. For k = 0, 1, 2, 3, 4, this gives the primes 3, 5, 17, 257, and 65537. 2^2^5 + 1 is not prime, however, and nobody has ever discovered another prime of this form, but nobody has ever proved that there aren't any more either. Looking just at the numbers from 3 to 20, we can build the following table of which regular polygons can and can't be constructed: 3 yes 3 = 2^2^0 + 1 4 yes 4 = 2^2 5 yes 5 = 2^2^1 + 1 6 yes 6 = 2 * 3 7 no 8 yes 8 = 2^3 9 no 10 yes 10 = 2 * 5 11 no 12 yes 12 = 2^2 * 3 13 no 14 no 15 yes 15 = 3 * 5 16 yes 16 = 2^4 17 yes 17 = 2^2^2 + 1 18 no 19 no 20 yes 20 = 2^2 * 5 Gauss was the first to discover that a regular polygon with 17 sides could be constructed. The construction of a regular polygon with 15 sides was known to the ancient Greeks. For more about constructable regular polygons, see the Dr. Math FAQ: http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html -Doctors Wilkinson and Sarah, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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