11 Jun
2003
11 Jun
'03
1:04 a.m.
At 02:29 AM 6/11/03, David Wilson wrote:
You wrote:
If there is an equilateral polygon with vertices in Q^2, it can be scaled so that its vertices are in Z^2.
If such a polygon has sides of even length, they can be halved (as any Pythagorean triangle with an even hypotenuse has even legs).
What if the sides are irrational?
Oops, I skipped a step. If the sides are irrational, we can scale and rotate to make them integers. Imagine Z^2 to be the Gaussian integers, and suppose one vertex is at 0 and the next is a + b i. Multiply all the vertices by a - b i; they remain Gaussian integers, and the side lengths are now a^2 + b^2 instead of sqrt(a^2 + b^2). -- Fred W. Helenius <fredh@ix.netcom.com>