Coxeter pairs, Ammann patterns, and Penrose-like tilings

Latham Boyle, Paul J. Steinhardt

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We identify a precise geometric relationship between (i) certain natural pairs of irreducible reflection groups ("Coxeter pairs"), (ii) self-similar quasicrystalline patterns formed by superposing sets of 1D quasiperiodically spaced lines, planes or hyperplanes ("Ammann patterns"), and (iii) the tilings dual to these patterns ("Penrose-like tilings"). We use this relationship to obtain all irreducible Ammann patterns and their dual Penrose-like tilings, along with their key properties in a simple, systematic and unified way, expanding the number of known examples from four to infinity. For each symmetry, we identify the minimal Ammann patterns (those composed of the fewest 1d quasiperiodic sets) and construct the associated Penrose-like tilings: 11 in 2D, 9 in 3D, and one in 4D. These include the original Penrose tiling, the four other previously known Penrose-like tilings, and sixteen that are new. We also complete the enumeration of the quasicrystallographic space groups corresponding to the irreducible noncrystallographic reflection groups, by showing that there is a unique such space group in 4D (with nothing beyond 4D).

Original languageEnglish (US)
Article number144113
JournalPhysical Review B
Issue number14
StatePublished - Oct 1 2022

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


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