HBS Tilings Extended: State of the Art and Novel Observations

TL;DR

This paper reviews the state of the art in HBS Penrose tilings, introduces new decorated tiles and substitution properties, and presents novel tilesets like Star and Gemstones, advancing understanding of aperiodic tilings.

Contribution

It provides a comprehensive synthesis of HBS tilings, introduces new decorated tiles, substitution rules, and novel tilesets such as Star and Gemstones.

Findings

Explicit substitution rules for new tilesets

Introduction of the Star tileset with three star variants

Development of the Gemstones tileset derived from Star tiles

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.