James Propp <jamespropp@gmail.com> wrote:
Here's a beautiful essay by Joel Hamkins that handles 5-gons, 6-gons, 7-gons, etc. with a uniform argument and then settles the case of 3-gons by appealing to the fact about 6-gons.
http://jdh.hamkins.org/no-regular-polygons-in-the-integer-lattice/
Clever. But I did have to really work at it to makes sense of it since Safari rendered all the text like this: The same argument works with larger regular polygons. The main point to realize is that for all regular [Math Processing Error] n-gons, where [Math Processing Error] n>4, when you construct the perpendicular on one of the sides, the resulting point is strictly inside the original polygon, and this is why the resulting regular [Math Processing Error] n-gon is strictly smaller than the original. This completes the proof for all [Math Processing Error] n-gons for [Math Processing Error] n>4. Any idea why? 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.