Geometric realizations of two dimensional substitutive tilings

TL;DR

This paper explores 2-dimensional topological substitutions for tilings in Euclidean and hyperbolic planes, proving the non-existence of primitive hyperbolic tilings and providing examples of non-primitive ones.

Contribution

It introduces the concept of 2D topological substitutions for tilings and establishes fundamental limitations for primitive hyperbolic tilings, with explicit examples of non-primitive cases.

Findings

No primitive substitutive tilings of the hyperbolic plane exist.

An example of a non-primitive substitutive tiling of the hyperbolic plane is provided.

The framework extends understanding of tiling substitutions in different geometries.

Abstract

We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove that there is no primitive substitutive tiling of the hyperbolic plane $\mathbb{H}^2$. However, we give an example of substitutive tiling of $\Hyp^2$ which is non-primitive.