Wouter writes: << when building a 3D set of points using a simple rule, I accidentally found a subset that seems to approach icosahedral symmetry. Why 'approach' ? because all points in my set are rationals, and those can never be icosahedral since they lack the necessary Sqrt[5] irrationality. The procedure is simple enough: start from the tetrahedron (zeroth generation) {0, 0, 0}, {1, 1, 0}, {1, 0, 1}, {0, 1, 1} and on each triangular face, add the point to form the fourth vertex of a new tetrahedron: . . . . . . . . . with the n-th generation having denominators 3^n, and the number of points tripling each generation. The quasi-icosahedron becomes apparent when selecting all points at Sqrt[2] from the origin. . . . . . . I didn't succeed in understanding where the underlying connection to their pseudo-icosahedral grouping comes from. You got to plot them to believe it
Interesting, but I don't see how this procedure, whose input is solely the vertices of a regular tetrahedron, can have anything but the original tetrahedral symmetry at each stage and therefore in the limit as well. (E.g., given a regular tetrahedron, one can find a regular icosahedron inscribed in it -- each of 4 faces of the icosahedron is a portion of one face of the tetrahedron -- but the icosahedron is not determined uniquely.) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele