A Recognizable Substitution Rule for a 10-fold Symmetric Rhomb Tiling

TL;DR

This paper introduces a recognizable substitution rule for a 10-fold symmetric rhomb tiling, enabling unique hierarchical reconstruction and linking it to Penrose tilings and pentagrid construction.

Contribution

It provides the first explicit, recognizable substitution rule for a 10-fold symmetric rhomb tiling with a detailed relation to Penrose tilings.

Findings

Substitution rule with inflation factor phi^3 established

Tiling's hierarchical structure is uniquely recoverable

Connection to pentagrid construction demonstrated

Abstract

We present a substitution rule for a rhomb tiling with 10-fold rotational symmetry. The tiling is closely related to the Penrose rhomb tilings and can be obtained from the pentagrid construction. We introduce a finite set of marked prototiles and describe an explicit substitution rule with inflation factor phi^3. Our main result is that the substitution is recognizable, so that the hierarchical structure of the tiling can be uniquely recovered from local configurations. Finally, we describe the relation between the tiling and the pentagrid construction.