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.