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.