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We identify a precise geometric relationship between: (i) certain naturalpairs of irreducible reflection groups (``Coxeter pairs"); (ii) self-similarquasicrystalline patterns formed by superposing sets of 1Dquasi-periodically-spaced lines, planes or hyper-planes (``Ammann patterns");and (iii) the tilings dual to these patterns (``Penrose-like tilings"). We usethis relationship to obtain all irreducible Ammann patterns and their dualPenrose-like tilings, along with their key properties in a simple, systematicand unified way, expanding the number of known examples from four to infinity.For each symmetry, we identify the minimal Ammann patterns (those composed ofthe fewest 1d quasiperiodic sets) and construct the associated Penrose-liketilings: 11 in 2D, 9 in 3D and one in 4D. These include the original Penrosetiling, the four other previously known Penrose-like tilings, and sixteen thatare new. We also complete the enumeration of the quasicrystallographic spacegroups corresponding to the irreducible non-crystallographic reflection groups,by showing that there is a unique such space group in 4D (with nothing beyond4D).

Coxeter Pairs, Ammann Patterns and Penrose-like Tilings