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

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.

Figures (5)

• Figure 1: A patch of the Seabed tiling.
• Figure 2: The six rhomb prototiles with edge markings.
• Figure 3: An equivalent representation of the prototiles with graphical markings.
• Figure 4: Substitution rule for the six prototiles. Each prototile (top) is replaced by the corresponding patch (bottom).
• Figure 5: The pentagrid construction.

Theorems & Definitions (14)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem