The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Math Topics » geometry.pre-college

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
John Conway

Posts: 2,238
Registered: 12/3/04
Re: 11-gon
Posted: Oct 31, 1994 5:58 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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

John Conway

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.