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
quasi-periodically-spaced lines, planes or hyper-planes (“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 non-crystallographic reflection groups,
by showing that there is a unique such space group in 4D (with nothing beyond

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