The permutahedron of order 5

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The permutahedron is the Cayley graph of the symmetric group S5 generated by the nearest-neighbour swaps (12), (23), (34) and (45). The permutation p is represented as the point p-1 in R5. All permutations lie on a 3-sphere, and this is projected onto R3.
Explanation for bell-ringers: Each vertex (corner) represents a doubles row. The coloured lines connect rows related by single changes: 345, 145, 125 and 123. Here are some methods.