A Challenging 7-Fold Tiling Puzzle

Abstract

A quasiperiodic 7-fold rhombic tiling is constructed with an iterative substitution scheme. The inflation factor is 5.04892..., the square of the longer diagonal of a regular heptagon. There are many substitutions possible that fill larger similar tiles with three base shapes but finding the matching rules (how the larger tiles fit together to make even larger tiles) turns into a challenging puzzle. At the end, a new solution is found with seven substitution rules. The relationship to a higher-order Fibonacci series is explored. The tiling is presented with a celestial theme reminiscent of a starry night.