2. Is it even possible for anything drawn on an integer grid to be affine to a Penrose tiling? Related question: for which N, can the vertices of a regular N-gon be affined to distinct integer coordinates?
--Answer: Theorem: it is impossible to affine a regular N-gon to get distinct integer coordinates for its vertices, unless N=2,3,4,6. Proof idea: (A) The integer linear combinations of the vertex vectors of a regular N-gon, form a dense set. (Except for the Ns listed.) (B) The integer linear combinations of a finite set of points with distinct integer coordinates, form a set of density <=1, which indeed is necessarily periodic (period arises from LCM). We know (B) does not hold for any N besides those listed because of 2D crystallography. QED. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)