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Topic: 11-gon
Replies: 15   Last Post: Mar 24, 2008 9:35 AM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: 11-gon
Posted: Oct 31, 1994 5:58 PM
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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 non-constructible regular polygons. It is in
fact true that if you are allowed an angle-trisector in addition to the
usual Euclidean tools, then you can construct all the regular p-gons
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 n-gons for n = 3,5,7,9,13,17 by applying the Galois-theory
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 angle-quinquesector. I wrote one down, but it was
very complicated. I also wrote down a construction for the regular
19-gon 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 7-gon : if anyone's
interested, I'll try to think up a way to convey my construction be
email.

John Conway





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