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Why there are 63 Types

Theorem : There are exactly 63 different (monochrome) wallpaper types. That is, there are exactly 63 equivalence classes of homomorphisms from E to the group {1,-1}, where E is isomorphic to one of the 17 wallpaper groups.

For each of the wallpaper groups we need to count the different possible homomorphisms to the group {1,-1} with the restriction that the kernel of the homomorphism contains two independent translations.

We list each group and the number of distinct homomorphisms for that group. To show that our list is exhaustive, we provide a table for each group (linked to its name below) with every possible homomorphism, indicating when any two are equivalent. For most types, there is an image of a wallpaper function having that type.

p1: 2 pmm: 6 cm: 4 p4: 3 p3: 1
p2: 3 pmg: 6 cmm: 6 p4g: 4 p31m: 2
pg: 3 pgg: 3 p4m: 6 p3m1: 2
pm: 6 p6: 2
p6m: 4

Proof:

2 + 3 + 3 + 6 + 6 + 6 + 3 + 4 + 6 + 3 + 4 + 6 + 1 + 2 + 2 + 2 + 4 = 63



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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 08 Jul 1998 --- Last modified: Sep 30, 2003 9:58:09 AM
Comments to: CVM@maa.org