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To begin with, there are the five "Platonic solids" beloved of geometers: the cube, the tetrahedron, the octahedron, the dodecahedron and the icosahedron. But why stop at the 3 dimensions of ordinary space? Alicia Boole Stott has devoted her life to finding regular solids in dimension 4... and she's found them! A journey into unsuspected mathematical regions.