Re: [math-fun] Odd-sided equilateral integer polygons
Also regular hexagons, e.g. (1 0 -1) (0 1 -1) (-1 1 0) (-1 0 1) (0 -1 1) (1 -1 0) —Dan ----- Duh: there's a triangle joining (1,0,0), (0,1,0), and (0,0,1). Jim Propp On Sunday, August 13, 2017, James Propp <jamespropp@gmail.com> wrote:
What happens in Z^3?
We can still checkerboard-color the lattice, and i^2+j^2+k^2 has the same parity as i+j+k. When i+j+k is odd, the proof goes through; but when i+j+k is even, there's no obvious analogue of rotating by 45 degrees and shrinking by sqrt(2)
And a regular tetrahedron as well: (0,0,0), (0,1,1), (1,0,1), (1,1,0). -- Gene On Sunday, August 13, 2017, 9:04:41 PM PDT, Dan Asimov <dasimov@earthlink.net> wrote: Also regular hexagons, e.g. (1 0 -1) (0 1 -1) (-1 1 0) (-1 0 1) (0 -1 1) (1 -1 0) —Dan ----- Duh: there's a triangle joining (1,0,0), (0,1,0), and (0,0,1). Jim Propp On Sunday, August 13, 2017, James Propp <jamespropp@gmail.com> wrote:
What happens in Z^3?
We can still checkerboard-color the lattice, and i^2+j^2+k^2 has the same parity as i+j+k. When i+j+k is odd, the proof goes through; but when i+j+k is even, there's no obvious analogue of rotating by 45 degrees and shrinking by sqrt(2)
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Eugene Salamin