Symmetry properties of Penrose type tilings

Abstract

The Penrose tiling is directly related to the atomic structure of certain decagonal quasicrystals and, despite its aperiodicity, is highly symmetric. It is known that the numbers 1, − τ, (− τ)², (− τ)³, …, where , are scaling factors of the Penrose tiling. We show that the set of scaling factors is much larger, and for most of them the number of the corresponding inflation centres is infinite.

Keywords:

• Penrose tiling
• quasicrystal
• self-similarities
• inflation factor
• strip projection method

Acknowledgements

The author is grateful to one of the referees for some very useful suggestions. This research was supported by the grant CEx05-D11-03.