It is known that both the rhombus and kite-and-dart Penrose tilings are three-colourable, from
- Babilon, Robert. “3-colourability of Penrose kite-and-dart tilings.” Discrete Mathematics 235, no. 1-3 (2001): 137-143.
- Sibley, Tom, and Stan Wagon. “Rhombic Penrose tilings can be 3-colored.” The American Mathematical Monthly 107, no. 3 (2000): 251-253.
Consider a three-colouring of a Penrose tiling using Red, Green, and Blue. I would like to find colourings that maximize the ratio of Red tiles to non-Red tiles, and to calculate that maximum (or supremum, if the maximum does not exist).
My motivations for this question are artistic, rather than mathematical. I have been experimenting with using aperiodic tilings in art, and I would imagine such an “unbalanced” colouring of a Penrose tiling would be visually interesting.