Talk:Uniform tiling

This is a quick stub, collecting lots of related elements of the uniform tilings but not organized very usefully. It'll hopefully it'll be slowly improved soon. Tom Ruen (talk) 01:50, 17 July 2008 (UTC)

Uniform tessellation
How does a uniform tiling differ from a uniform tessellation? These are separate articles, but it's not clear to me what the difference is. — Nonenmac (talk) 14:10, 31 July 2008 (UTC)

Euclidean analogues

 * Section moved from Talk:Hemipolyhedron

The obvious Euclidean analogues are the nonconvex tilings involving apeirogons: as Coxeter considered and enumerated many of these, they should be fair game for Wikipedia. (There are also higher-dimensional analogues among spherical and Euclidean tessellations, but those are not fair game AFAIK as they are absent from reliable sources.) Double sharp (talk) 04:49, 5 September 2018 (UTC)
 * Like these? Uniform_tiling Tom Ruen (talk) 22:04, 5 September 2018 (UTC)
 * Yes, exactly. I think we should have some pictures of those tilings there as well, as opposed to just showing their vertex figures. (I agree that overlapping geometry makes things difficult, especially when all faces are coplanar, but we could use colour around one vertex and leave the rest white.) Maybe I should make a companion to List of uniform polyhedra by Schwarz triangle as well! Double sharp (talk) 05:16, 6 September 2018 (UTC)
 * Some star tiling images are here: Tom Ruen (talk) 20:24, 6 September 2018 (UTC)
 * Those pictures are nice, but I think they are achieving clarity by sacrificing truth. Consider 3/2 6 | 6. Now this is clearly analogous to the small dodecicosidodecahedron 3/2 5 | 5. The convex hull of that polyhedron is the rhombicosidodecahedron, which contains pentagonal cupolae. Since the small dodecicosidodecahedron has both the base decagons of those cupolae and the top pentagons as faces, these decagons and pentagons are clearly not in the same plane, and the triangles between them must angle upwards out of the plane of a decagon to connect them. But if you drew the small dodecicosidodecahedron as a spherical tiling, everything would overlap exactly on the sphere's surface: the pentagons and triangles would be completely swallowed up by the decagons. The situation with 3/2 6 | 6 (which we could analogously call the small hexatrihexagonal tiling) should look like the second, and not the first. But the drawing is making it look more like the first, because the second would be much too confusing, like the pictures you give below. So I'm not sure how to solve this problem, though at least the hemis containing unfilled apeirogonal faces look clearer. Double sharp (talk) 04:40, 7 September 2018 (UTC)

Here's some quick images (reflective ones), with solid faces and white background, so too much overlappping on later ones to clearly see structure. Tom Ruen (talk) 21:04, 6 September 2018 (UTC) I added wireframe (vertex/edge) versions. Tom Ruen (talk) 21:58, 6 September 2018 (UTC)

Could we highlight the edge sequences of some apeirogons? It would be good to show by analogy one apeirogonal equator of the original r{4,4}, r{6,3}, and r(3,3,3). Double sharp (talk) 04:55, 7 September 2018 (UTC)

It seems to me that the hyperbolic analogues ought to have pseudogons as faces instead, as when {p,q} is hyperbolic, computing the equator {h} of r{p,q} by the formula cos2 (&pi;/h) = cos2 (&pi;/p) + cos2 (&pi;/q) yields imaginary h. This h is of course also the Petrie polygon of {p,q} and {q,p}. I don't remember seeing them at the website you linked, but presumably these should exist (e.g. {3,2n} with only half the triangles, generalising the tetrahemihexahedron; and r{p,q} with only the p-gonal or only the q-gonal faces, generalising the other hemipolyhedron pairs). These pseudogons would have to have the right curvature to lie flat along a line rather than be inscribable in some hypercycle, like Euclidean apeirogons (hyperbolic apeirogons are inscribed in horocycles). But this would be OR in any case. Double sharp (talk) 08:49, 7 September 2018 (UTC)
 * p.s. Some of these seem related to the Regular complex apeirogons. I made a summary showing pattern and one vertex figure there. File:Complex_apeirogon_chart2.png. There could be more similarities, but there's not supposed to be any structural order to 1-dimensional polytope p-ons so p/q-ons don't exist. Tom Ruen (talk) 13:16, 7 September 2018 (UTC)
 * How about moving this discussion to Talk:Uniform tiling? Tom Ruen (talk) 13:22, 7 September 2018 (UTC)
 * Sure, that sounds like a good idea! Double sharp (talk) 15:31, 7 September 2018 (UTC)

Perhaps images like those at Richard Klitzing's site (mostly from Jim McNeill's site) would be better for showing the overlapping geometry, only colouring around one vertex? Double sharp (talk) 15:03, 9 September 2018 (UTC)
 * I've uploaded McNeill's images (including the two snubs at the end) above. I didn't have the patience to use the vertex configurations, and these tilings don't seem to have standard names, so I just used the Bowers acronyms as file names. Double sharp (talk) 15:00, 15 September 2018 (UTC)
 * P.S. | 2 4/3 4/3 is actually reflexible. Double sharp (talk) 15:09, 15 September 2018 (UTC)
 * ...and I've replaced the pictures in the article. Now to add a section on Euclidean tilings to List of uniform polyhedra by Schwarz triangle, just like Coxeter et al. do in their famous paper! Double sharp (talk) 15:13, 15 September 2018 (UTC)

Uniform tilings using star polygons
Note that the recently added list of uniform tilings using star polygons (interpreted as nonconvex 2n-gons) is incomplete. There are two other examples, missed in Tilings and Patterns, in my article in Eureka 56. Joseph Myers (talk) 20:42, 22 September 2020 (UTC)