A Challenging 7-Fold Tiling Puzzle

TL;DR

This paper presents a novel 7-fold quasiperiodic tiling constructed via an iterative substitution scheme, exploring its mathematical properties and solving a complex matching rules puzzle, with connections to higher-order Fibonacci sequences.

Contribution

It introduces a new 7-fold tiling with specific substitution rules and investigates its mathematical structure and relationships to Fibonacci sequences.

Findings

A new 7-fold quasiperiodic tiling is constructed.

Seven substitution rules are identified for the tiling.

The tiling relates to higher-order Fibonacci series.

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.