(some notes for collaborative study)
is this still unproven? ->
A Penrose tiling can, of course, always be colored with four colors so that no two tiles of the same color share a common edge. Can it always be colored with three? It can be shown, Conway said, from the local isomorphism theorem, that if any Penrose tiling is three-colorable, all are, but so far no one has proved that any infinite Penrose tiling is three-colorable.