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)

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)

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)