Randomness and Topological Invariants in Pentagonal Tiling Spaces

Abstract

We analyze substitution tiling spaces with fivefold symmetry. In the substitution process, the introduction of randomness can be done by means of two methods which may be combined: composition of inflation rules for a given prototile set and tile rearrangements. The configurational entropy of the random substitution process is computed in the case of prototile subdivision followed by tile rearrangement. When aperiodic tilings are studied from the point of view of dynamical systems, rather than treating a single one, a collection of them is considered. Tiling spaces are defined for deterministic substitutions, which can be seen as the set of tilings that locally look like translates of a given tiling. Čech cohomology groups are the simplest topological invariants of such spaces. The cohomologies of two deterministic pentagonal tiling spaces are studied.

A Computer Search for Planar Substitution Tilings with $$n$$ n -Fold Rotational Symmetry

“…For these prototiles, the smallest inflation factor is a 2 , the golden ratio. Substitutions with this inflation factor have been described elsewhere (see [19] and [11]); these can be combined as multi-substitutions, but not as random substitutions-although see [11] for a description of a slightly different randomization strategy, called tile rearrangement.…”

“…(Some terms mentioned here, such as "prototile," will be defined formally in Section 1.3.) These fall into two classes: the multi-substitution tilings (see [5], [4], [6], [10], [11], and [20]), that are obtained by choosing a substitution for each hierarchical level and applying it to all tiles at that level; and the random substitution tilings (see [12], [17], [18], and [2]), that are obtained by making separate choices of substitution for each tile at each hierarchical level. While it is easy to find examples of such families of substitutions in one dimension, in two dimensions it is harder.…”

Rotationally symmetric tilings with convex pentagons and hexagons

“…For the non-periodic case, which is studied here, we will see that we can go further. From related work (e. g. [11], [12]) tilings are known with arbitrary rotational symmetry but they are generated by rhombs or triangles as prototiles. Here we define a class of pentagons, from which one can generate the proposed symmetry types.…”