Yeah, that's subsumed by my inner product argument. Also, the vertices of a pentagon are a subset of those of the dodecahedron and icosahedron, so they follow as trivial corollaries. Sincerely, Adam P. Goucher
----- Original Message ----- From: Warren D Smith Sent: 05/26/13 07:23 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Embedding with integer coordinates
If your thing has got two interpoint distances A,B such that (A/B)^2 is irrational, then its points cannot be be embedded in any finite dimension with integer coordinates.
Regular pentagon, (A/B)^2=2.6180339887498948482 = g+1 where g=golden ratio where B=side, A=2-apart vertex sep
icosahedron (including center as point) (A/B)^2 = sin(2*pi/5)^2 = (g/2)^2 + 1/4 =0.9045084971874737120
dodecahedron (ditto) (A/B)^2 = g^2 * (3/4) = 1.96352549156242113610 where A=radius, B=side
all easily seen to be irrational, so none can be embedded.
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