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June 28

Restricted Wang tiles

Has anyone every investigated what happens if you restrict Wang tiles to only include those where two given sides (say north and west) have the same color? For example, are there still sets which tile the plane aperiodically but not periodically? They have a symmetric geometric interpretation if you rotate the plane a bit and use a hexagonal instead of a square lattice. For n colors, instead of 2^4ⁿ possible sets there are only 2^3ⁿ, which seems like a lot less. At first I thought that problems involving this simplified form would be solvable in polynomial time, but now I'm starting to believe they have the same complexity as full Wang tiles. I'm not really expecting an answer here, but you never know until you ask. --RDBury (talk) 15:24, 28 June 2021 (UTC)[reply]

This contribution by Karel Čulík to a festschrift for Arto Salomaa shows how to simulate square Wang tiles by replacing each by a configuration of four hexagonal Wang tiles. I don't know if this has any bearing on your question, though. --Lambiam 19:27, 28 June 2021 (UTC)[reply]

Thanks. Unfortunately the the geometric interpretation is as simple as triangular tiles. More like triangular trihexes where overlapping hexagons are required to be the same rather than adjacent edges. You can probably see the pattern by finding the unique plane tiling by the following tiles:

  0      1      2
0   1, 1   2, 2   0
  2      0      1

It's interesting that triangular tiles have the same complexity though. It is possible to interpret this as a subset of hexanogonal tiles, specifically by

              a
  a       a       b
a   b -> 
  c       c       b
              c

but I'm not sure that helps. --RDBury (talk) 01:09, 29 June 2021 (UTC)[reply]

identical 4 & 5 face covers for icosahedron?

For the regular Polyhedra (with n sides), I've managed to work out that the faces can *always* be split into q equal connected pieces if q divides n *except* for 4 and 5 face pieces on the icosahedra. Can someone help me figure those out (or let me know that there is no such?) For example, a Dodecahedron can be covered by 6 identical 2 pentagon units, 4 identical 3 pentagon units, 3 identical 4 pentagon units and 2 identical 6 pentagon units.Naraht (talk) 18:43, 28 June 2021 (UTC)[reply]

A 4 face piece tiling isn't too hard to find. Pick a vertex and its opposite as "poles", then draw 5 zig-zag longitude lines connecting the poles. I'm not sure about 5 face pieces. --RDBury (talk) 01:28, 29 June 2021 (UTC)[reply]

I found a 5 face piece tiling, in fact several, as well though it's a bit more complicated to describe. I'll work finding a way to specify it. --RDBury (talk) 01:53, 29 June 2021 (UTC)[reply]

Label the vertices with their coordinates all possible cyclic permutations and sign-flips of (ϕ, 1, 0). One piece has boundary vertices (in order) (ϕ, 1, 0), (ϕ, −1, 0), (0, −ϕ, 1), (1, 0, ϕ), (−1, 0, ϕ), (0, ϕ, 1), (0, ϕ, -1). (The shape is third on the list of 5 triangle Polyiamonds in the table under Counting.) The remaining three pieces are obtained by applying an even number of sign flips to the first piece. --RDBury (talk) 02:21, 29 June 2021 (UTC)[reply]