Affine symmetric group

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Flat, straight-edged shapes (like triangles) or 3D ones (like pyramids) have only a finite number of symmetries - ways to reflect or rotate it where it looks the same. In contrast, the affine symmetric group is a way to mathematically describe all the symmetries possible when an infinitely large flat surface is covered by triangular tiles. As with many subjects in mathematics, it can also be thought of in a number ways: for example, it also describes the symmetries of the infinitely long number line, or the possible arrangements of all whole numbers (..., −2, −1, 0, 1, 2, ...) with certain repetitive patterns. As a result, studying the affine symmetric group extends the study of symmetries of polyhedra or of groups of permutations to the infinite case. It also connects several topics in mathematics that were originally studied for independent reasons, ranging from complex reflection groups to juggling sequences.