HBS and star tilings: State of the art and novel observations

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. However, 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.