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Re: 11gon
Posted:
Oct 31, 1994 5:58 PM


Of course Gauss's construction of the regular heptadecagon and his general theory of which regular polygons are constructible is one of the great triumphs of mathematics, and is particularly interesting as relating number theory to Euclidean geometry. Let's continue to celebrate. Gauss himself took pride in this all his life, and wanted it engraved on his tombstone. It wasn't, but it was indeed put on the memorial to him in Braunsweig (or, since we've been discussing national spellings, Brunswick as we once called it).
Someone asks about the nonconstructible regular polygons. It is in fact true that if you are allowed an angletrisector in addition to the usual Euclidean tools, then you can construct all the regular pgons for which p is a prime number of the form 2^a.2^b + 1. This keeps on being rediscovered for the case p = 7.
I wrote down what I think are particularly nice and uniform constructions for the regular ngons for n = 3,5,7,9,13,17 by applying the Galoistheory by which this is proved DIRECTLY to the geometrical problem. In particular, this gives a construction for the regular heptagon that is much simpler and neater than those people usually get, by just solving the relevant cubic trigonometrically.
The theorem generalizes as one would expect  so for instance there is a construction of a regular hendecagon using ruler and compasses together with an anglequinquesector. I wrote one down, but it was very complicated. I also wrote down a construction for the regular 19gon using an angle trisector twice, but this was also too complicated to keep. This is something I hope to get back to sometime  there ought to be fairly simple constructions that I'm missing.
Andrew Gleason has an old paper constructing the 7gon : if anyone's interested, I'll try to think up a way to convey my construction be email.
John Conway



