TY - JOUR
T1 - Coxeter pairs, Ammann patterns, and Penrose-like tilings
AU - Boyle, Latham
AU - Steinhardt, Paul J.
N1 - Funding Information:
We thank Joshua Socolar for a helpful private communication first drawing our attention to the 12-fold “A2” Ammann pattern and Penrose tiling, as well as the tenfold Ammann trio. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities. P.J.S. was supported in part by the Princeton University Innovation Fund for New Ideas in the Natural Sciences.
Publisher Copyright:
© 2022 American Physical Society.
PY - 2022/10/1
Y1 - 2022/10/1
N2 - 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).
AB - 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).
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U2 - 10.1103/PhysRevB.106.144113
DO - 10.1103/PhysRevB.106.144113
M3 - Article
AN - SCOPUS:85141448044
SN - 2469-9950
VL - 106
JO - Physical Review B
JF - Physical Review B
IS - 14
M1 - 144113
ER -