Very nice! Jim On Sun, Aug 13, 2017 at 9:13 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
What about equilateral polygons in the integer grid? A dodecagon is possible if it's in the form of a Greek cross (aka X-pentomino). Are all even-N-gons possible? I suspect that no odd-N-gon is possible. I'm searching for a proof.
Suppose we have an equilateral polygon with vertices in the Gaussian integers and common side-length sqrt(L), where L is an integer. Without loss of generality, assume that one of the vertices is at the origin.
Now, if L is even, then for any pair of adjacent vertices v, w in Z[i], we have that (v - w) is divisible by 1 + i. Hence, by induction, every vertex is divisible by 1 + i, so we can divide throughout and obtain an equilateral polygon with common side-length sqrt(L/2).
Hence, we can assume without loss of generality that L is odd. Now colour the vertices of the integer grid red and blue in a checkerboard fashion. Every pair of adjacent vertices in an equilateral polygon of common side length L have opposite colours, so no odd-sided equilateral N-gons can exist. QED.
Best wishes,
Adam P. Goucher
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun