Date: Dec 21, 1994 9:52 AM
Author: John Conway
Subject: Re: Polygons with compass & straight edge
It's a VERY famous theorem of Gauss that the only regular polygons

with a prime number of sides that can be constructed with straightedge

and compass are those for which the prime is one of the Fermat primes

3, 5, 17, 257, 65537, ...

(that is, primes of the form 2^n + 1). Nobody knows if there are

any Fermat primes larger than 65537.

The only constructible regular polygons with an odd number

of sides are those for which this number is a product of distinct

Fermat primes (so for instance 15 = 3 times 5, 51 = 3 times 17),

and the only ones with an even number of sides are those obtained

by repeatedly doubling these numbers (including 1), thus:-

(1,2), 4, 8, 16, 32, 64, ...

3, 6, 12, 24, 48, ...

5, 10, 20, 40, 80, ...

15, 30, 60, ...

17, 34, 68,...

51, ...

85,...

I see that I've listed everything below 96 explicitly.

Some people might like the following little observation.

Write out the Pascal triangle modulo 2 :-

1

1 1

1 0 1

1 0 0 1

1 1 0 1 1

1 0 1 1 0 1

1 1 1 1 1 1 1

1 0 0 0 0 0 0 1

.................

then by reading the first 31 rows as the binary expansions of

numbers you get

1, 3, 5, 15, 17, 51, 85, 255, 257, ...

which give the first few odd-sided constructible polygons (and

very probably all there are).

John Conway