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HBS Tilings Extended: State of the Art and Novel Observations

Carole Porrier

TL;DR

The paper addresses consolidating and extending the study of HBS tilings by introducing a Star tileset with three star variants, yielding five prototiles that reproduce strict HBS tilings when decorations are removed. It proposes new decorations and forcing rules, including explicit $\varphi$- and $\varphi^{2}$-decompositions and substitutions that connect HBS, Star, and Gemstones tilings. It analyzes mutual local derivability with Penrose tilings, detailing local mappings and the role of Ammann-bar decorations, and discusses tile frequencies, empires, and kingdoms that constrain local configurations. Overall, the work unifies HBS tilings with Penrose tilings, expands the tile-sets with new substitution structures, and provides a practical framework for quasicrystal-inspired tilings.

Abstract

Penrose tilings are the most famous aperiodic tilings, and they have been studied extensively. In particular, patterns composed with hexagons (H), boats (B) and stars (S) were soon exhibited and many physicists published on what they later called HBS tilings, but no article or book combines all we know about them. This work is done here, before introducing new decorations and properties including explicit substitutions. For the latter, the star comes in three versions so we have 5 prototiles in what we call the Star tileset. Yet this set yields exactly the strict HBS tilings formed using 3 tiles decorated with either the usual decorations (arrows) or Ammann bar markings for instance. Another new tileset called Gemstones is also presented, derived from the Star tileset.

HBS Tilings Extended: State of the Art and Novel Observations

TL;DR

The paper addresses consolidating and extending the study of HBS tilings by introducing a Star tileset with three star variants, yielding five prototiles that reproduce strict HBS tilings when decorations are removed. It proposes new decorations and forcing rules, including explicit - and -decompositions and substitutions that connect HBS, Star, and Gemstones tilings. It analyzes mutual local derivability with Penrose tilings, detailing local mappings and the role of Ammann-bar decorations, and discusses tile frequencies, empires, and kingdoms that constrain local configurations. Overall, the work unifies HBS tilings with Penrose tilings, expands the tile-sets with new substitution structures, and provides a practical framework for quasicrystal-inspired tilings.

Abstract

Penrose tilings are the most famous aperiodic tilings, and they have been studied extensively. In particular, patterns composed with hexagons (H), boats (B) and stars (S) were soon exhibited and many physicists published on what they later called HBS tilings, but no article or book combines all we know about them. This work is done here, before introducing new decorations and properties including explicit substitutions. For the latter, the star comes in three versions so we have 5 prototiles in what we call the Star tileset. Yet this set yields exactly the strict HBS tilings formed using 3 tiles decorated with either the usual decorations (arrows) or Ammann bar markings for instance. Another new tileset called Gemstones is also presented, derived from the Star tileset.
Paper Structure (4 sections, 2 equations, 15 figures)

This paper contains 4 sections, 2 equations, 15 figures.

Table of Contents

  1. Penrose tilings and mutual local derivability
  2. HBS tilings summarized
  3. The Star tileset
  4. Gemstones tileset

Figures (15)

  • Figure 1: Usual HBS tiles and their new, enriched version.
  • Figure 2: Penrose tilesets.
  • Figure 3: The seven vertex configurations in a Penrose tiling by kites and darts.
  • Figure 4: (a) From a P3 tiling, remove all edges with a double arrow and the vertices they point to. (b) From a P2 tiling, trace the axis of symmetry of each kite, with an arrow pointing to the wide angle, then erase all edges of kites and darts.
  • Figure 5: HBS tiles with (a) Ammann segments (arrows are omitted) and (b) their $\varphi^2$-decomposition.
  • ...and 10 more figures