Talk:List of isotoxal polyhedra and tilings

Euclidean tilings

The article lists five isotoxal Euclidean tilings : hexagonal, triangular, trihexagonal, rhombille, and square.

What about the rhombus tiling? Not the "rhombille", but simply a "tilted" square tiling. I know this has the same topology/combinatorics as the square tiling, but the symmetry group is different, which is important here because we are talking about transitivity.

Also, the rhombus tiling is vertex-, edge-, and face-transitive, but not regular [not flag-transitive]. Before I found this, I had the impression that a polyhedron or tiling was regular if and only if it had all three transitivities. (A single rhombus is not vertex-transitive, but the tiling is.) — Preceding unsigned comment added by Mr e man2017 (talk • contribs) 01:07, 18 August 2017 (UTC)[reply]

On second thought, maybe they were talking about combinatorial symmetries instead of geometric symmetries. That should have been stated clearly.

With geometric symmetries, there would be many more than 5 isotoxal tilings, because irregular tiles & angles are allowed, as long as it follows one of the 17 wallpaper groups. One example is made of equilateral triangles and 90°-150° hexagons, arranged in a trigyro (Conway's name) pattern. This has different geometric symmetry, but the same combinatorial symmetry as the trihexagonal tiling.

With combinatorial symmetries, there may be exactly 5 tilings. If this is indeed the case, then someone can edit the page to say so.

-- Mr e man2017 (talk) 06:52, 21 March 2018 (UTC)[reply]

There are actually more than 5 if you allow self-intersecting tilings (analogous to star polyhedra). Double sharp (talk) 08:01, 18 September 2018 (UTC)[reply]