ABSTRACT
We next show how to enumerate the p-fold colorings of plane patterns for any prime p. The new argument uses a fair amount of group theory. In group theorists’ slang, the group G of a Euclidean plane pattern has shape L.Q, meaning that it has a normal subgroup L with finite quotient Q. The group L ∼= C∞ × C∞ (“the lattice”) consists of the translations inG, and its quotient groupQ (“the point group”) is not necessarily realized as a subgroup. The argument works for a prime p that doesn’t divide the order ofQ (so, for p ≥ 5 in the cases ∗632, 632, ∗333, 3∗3, and 333, and for p ≥ 3 otherwise.) The missing threefold cases are precisely those that were done in the previous chapter.
