Talk:List of uniform polyhedra

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This is a first pass article, including the complete list of 75 uniform polyhedra, 11 uniform tessellations, and sampling of infinite sets of prism and antiprism.

Tom Ruen 00:01, 7 October 2005 (UTC)

I think it would make sense to include Skilling's great disnub dirhombidodecahedron [1] (Phil. Trans. A 278 (1975), pp 111-135). In view of the inclusion of tilings in this page and of uniform tilings with hollow tiles in Table 8 of Uniform Polyhedra (Phil. Trans. A 246 (1954), pp 401-450), there is a logical case for covering all the uniform tilings with hollow tiles, apeirogons and zigzags listed by Grünbaum, Miller and Shephard (in The Geometric Vein: The Coxeter Festschrift, pp 17-64), but that's a large collection, without proof of completeness, and if you include them then there's the question of infinite uniform polyhedra in three dimensions where I don't know of a good comprehensive list of the known polyhedra for any definition you might choose. Joseph Myers 19:42, 8 October 2005 (UTC)

Is completeness a requirement for listing a valid class of uniform polyhedra here? If so, it should be stated which such classes are being left out, otherwise it implicitly suggests they are not valid. Cutelyaware (talk) 05:11, 23 July 2015 (UTC)

I rewrote the intro a bit. It probably can be improved more. Tom Ruen (talk) 05:39, 23 July 2015 (UTC)

P.S. On "great disnub dirhombidodecahedron", very interesting. Going beyond 2 faces/edge is fun to consider, but I don't think you can call that a polyhedron. Maybe a "Polyhedron network" or something else since it would enclose multiple volumes. Such an ennumeration has a place in geometry, but not here, except as a link if such an article is writen. Tom Ruen 04:24, 15 October 2005 (UTC)

[More recently] I've been rethinking my rejection of this "polyhedron". Initially I rejected it as a true polyhedron for having more than 2 faces per edge. On RETHINKING, technically an even number of faces per edge might be considered as a topological polyhedron with geometrically overlapping edges. Well, I'll have to think further on this! Tom Ruen 20:42, 5 December 2005 (UTC)

It is quite valid for a polyhedron to contain coincident edges that do not interact. You just need to be specific about the topology. That still describes a simple manifold and is a very different situation from the idea of a polyhedron in which at least one edge is explicitly adjacent to more than two faces. Cutelyaware (talk) 05:11, 23 July 2015 (UTC)

Sure, its a matter of how you want to count, but all degeneracies are tricky, and you can understand why some want to exclude them. Tom Ruen (talk) 05:39, 23 July 2015 (UTC)

Vertex figure error?

The "vertex figure" for W79 looks like the polyhedron itself. ?? --Anton Sherwood 06:43, 7 January 2006 (UTC)

In the image for W114, the pentagrams look farther apart than in Wenninger's picture. --Anton Sherwood 07:06, 7 January 2006 (UTC)

--> Late note - I've corrected link for VF/W79 and uploaded correct image for W114. Tom Ruen 04:01, 1 February 2006 (UTC)

hyperbolica

• I don't fully understand the table, but it looks to me like the 2 7 | 3 entry was missing. I sketched one in. If it belongs, and if there is no image of the "bizarro football" to be found, I can cook one up. --Unzerlegbarkeit (talk) 16:56, 8 August 2008 (UTC)

• While I'm here, I threw in | 2 3 7 and 3 7 | 2, so I think all the 2-3-7 ones are there. To my surprise I guessed at an article name for small rhombitriheptagonal tiling, and it existed. I can cook up the other tiling. I'm not sure why tilings are included at all though. --Unzerlegbarkeit (talk) 18:41, 8 August 2008 (UTC)

There's infinitely MANY missing from the hyperbolic tilings. There's a newer article Uniform tilings in hyperbolic plane and Uniform tiling with a sampling of the hyperbolic tilings. They were added here to show the relations of vertex figures, but probably should be removed now, since the comparisons are given elsewhere as well. Tom Ruen (talk) 00:02, 10 August 2008 (UTC)

Any list of hyperbolic tilings ought (imho!) to include 433, 542 and 732, because each of these is minimal in the sense that reducing any of the numbers results in a tiling of E2 or S2. For the same reason I consider such a list sufficient for illustration; or do higher finite numbers bring more interesting properties? —Tamfang (talk) 22:44, 13 August 2008 (UTC)

I think we could have *433, *542 and *732 for hyperbolic. Possibly *832, but not higher. Professor M. Fiendish, Esq. 13:44, 6 September 2009 (UTC)

There is some point to higher numbers, though not on this page (anymore), because evens and odds behave differently in terms of alternation possibilities. So *832 is quite natural to include because it is double the symmetry of *433. The first all-evens case, *642, is also a natural inclusion to show all possibilities, and then you'd get *443 from that. Hmm, similarly *552 and *662 naturally arise from *542 and *642. Double sharp (talk) 10:45, 21 August 2022 (UTC)

Uniform duals

We seem to be missing a lot of dual polyhedra. As a start, I'll write up an article about the tetrahemihexacron. --Professor M. Fiendish 03:19, 23 August 2009 (UTC)

By the way, could Tomruen please get a picture of the tetrahemihexacron ready? Professor M. Fiendish 03:24, 23 August 2009 (UTC)

We also seem to be missing a lot of uniform dual images. Professor M. Fiendish 03:29, 23 August 2009 (UTC)

The hemi- forms are problematic for making duals because faces pass through the origin. The dual polyhedron construction inverts radial distances of faces centers from the model center, putting vertices at infinity! Magnus Wenninger's models make models with infinite prisms intersecting the origin, truncated to a finite distance. So this model looks sort of like 3D coordinate axes, not very interesting. Stella (software) can generate the images. Tom Ruen (talk) 04:51, 23 August 2009 (UTC)

ALSO, on adding individual articles, there's opinions already that the nonconvex uniform polyhedrons are not sufficiently notable for individual articles, so adding duals moves us even further into debate on this issue. At this point I'd say it's better to add a section to the uniform individual articles about the duals, if there's anything to be written about them. Tom Ruen (talk) 04:53, 23 August 2009 (UTC)

LASTLY! At the moment the nonconvex uniform polyhedron article itself is rather underwelming, merely lists them in groups by their vertex arrangements, but no clarity on their construction from Schwarz_triangles. I'm sure each one has many symmetry constructions, (with different face colorings), but I've not worked with Coxeter's original 1954 paper sufficiently to see what more can be extracted about their enumeration. I can email you a PDF of the paper if you like! Tom Ruen (talk) 04:57, 23 August 2009 (UTC)

I've went ahead and added some stubby articles. Mathworld has some images, which seem to look quite interesting. You've really enlightened me by telling me why there aren't pictures of the tetrahemihexacron and other related stuff here.

Some of the articles do contain a bit more substantial material: the hexahemioctacron is visually indistinct from the octahemioctacron. I'll continue adding more articles for now. If anyone objects, they can just redirect like tetrahemihexacron --> tetrahemihexahedron and add any information about the duals there.

I'm going to create the dual articles for now anyway. Professor M. Fiendish, Esq. 07:07, 23 August 2009 (UTC)

I saw the new articles Hemihedron and Hemicron, moved them to hemipolyhedron (Wenninger uses this term, somesources hyphenated) and hemipolycron (not used sources I have, but better parallel to first term) and expanded the content. Tom Ruen (talk) 22:22, 23 August 2009 (UTC)

OH MY EDITS!

I ADDED IN THE OCTAGRAMMIC PRISM BUT THEN THE PENTAGRAMMIC ANTIPRISM WAS GONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! PLEASE FIX IT!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! — Preceding unsigned comment added by 99.185.0.100 (talk) 21:00, 5 January 2014 (UTC)

You were missing a "|-" symbol for a new table row, so the antiprism rows ended up on the far right. Tom Ruen (talk) 22:23, 5 January 2014 (UTC)

density

How do you define density for hemipolyhedra? Since there are faces passing through the centre. Double sharp (talk) 13:41, 15 March 2014 (UTC)

According to Stella. ("U81", "W120" are not official designations for Skilling's figure: they were just so that the table would automatically sort it at the bottom. The U numbers beyond 75 are really references to prismatic families: U76 = convex prism, U77 = convex antiprism, U78 = nonconvex prism, U79 = nonconvex antiprism, U80 = crossed antiprism. These have been utterly unofficially been assigned W121 to W125 below for table sorting.)

U#	W#	Bowers	Density
1	1	Tet	1
6	3	Cube	1
5	2	Oct	1
23	5	Doe	1
22	4	Ike	1
2	6	Tut	1
7	11	Co	1
8	7	Toe	1
9	8	Tic	1
10	13	Sirco	1
11	15	Girco	1
12	17	Snic	1
24	12	Id	1
25	9	Ti	1
26	10	Tid	1
27	14	Srid	1
28	16	Grid	1
29	18	Snid	1
34	20	Sissid	3
35	21	Gad	3
52	22	Gissid	7
53	41	Gike	7
4	67	Thah	0
3	68	Oho	0
15	78	Cho	0
13	69	Socco	2
18	86	Sroh	0
19	92	Quith	7
21	103	Groh	0
14	77	Gocco	4
17	85	Querco	5
16	79	Cotco	4
20	93	Quitco	−1
49	89	Seihid	0
51	91	Sidhid	0
33	72	Saddid	2
39	74	Sird	0
66	104	Quit Gissid	13
58	97	Quit Sissid	9
63	101	Giddy	0
42	81	Gidditdid	4
48	88	Giid	6
45	84	Idtid	4
59	98	Quitdid	−3
68	108	Gaquatid	13
37	75	Tigid	3
55	95	Tiggy	7
65	102	Gidhei	0
36	73	Did	3
62	100	Sidhei	0
56	96	Ri	0
38	76	Raded	3
44	83	Ided	4
71	106	Geihid	0
54	94	Gid	7
70	107	Gidhid	0
47	87	Gidtid	6
30	70	Sidtid	2
41	80	Ditdid	−4
50	90	Siddy	0
31	71	Siid	2
43	82	Sidditdid	4
73	109	Gird	0
61	99	Gaddid	10
67	105	Qrid	13
32	110	Seside	2
40	111	Siddid	3
46	112	Sided	4
60	114	Isdid	9
57	113	Gosid	7
69	116	Gisid	13
74	117	Girsid	37
72	118	Sirsid	38
64	115	Gisdid	10
75	119	Gidrid	0
81	120	Gidisdrid	0
76	121	n-p	1
77	122	n-ap	1
78	123	n/d-p	d
79	124	n/d-ap	d
80	125	n/(n−d)-ap	n−d

Double sharp (talk) 06:36, 27 April 2014 (UTC)

Here's a link to Coxeter's famous paper

http://rsta.royalsocietypublishing.org/content/roypta/246/916/401.full.pdf Double sharp (talk) 01:56, 16 March 2015 (UTC)

I see linked from here [2]

Density sorting

I corrected some values, substituting the ones from Coxeter's paper. It sorts wrongly though (3, 37, 38, 4...). T_T

In particular, the hemipolyhedra and even-faced nonorientables don't really have a well-defined density, so I replaced them with "--". I think that's what's messing with the sorting.

A few other corrections, based on Coxeter's paper:

• Great truncated cuboctahedron (quitco) – density 1, not 7. We expect 7, as that is the density of the spherical triangle (2 3 4/3), but the vertex figure is an obtuse triangle. A distorted version with an acute vertex figure would indeed have density 7, but as we make the vertex figure's angle grow larger, the eight hexagonal faces pass through the centre and out the other side, and the density now becomes |7 − 8| = 1. Now it has the same density as the triangle (2 3 4) sharing the same great circles. You can see the effect of this in the small triangular holes in the polyhedron.
• Truncated dodecadodecahedron (quitdid) – density 3, not 9. This is essentially the same idea: (2 5/3 5) has density 9, but the vertex figure is obtuse: now we have twelve decagonal faces passing through the centre in the distortion, giving a density of |9 − 12| = 3, the same as (2 5/2 5) which shares the same great circles. Again, you get small triangular holes.
• Snubs – in three cases (great icosahedron, small retrosnub icosicosidodecahedron, great retrosnub icosidodecahedron) the vertex figure is so distorted that it becomes a pentagram/hexagram instead of a pentagon/hexagon. So now we need to subtract the number of snub faces from the density of the Schwarz triangle involved (same idea as the previous two cases), giving |5 − 12| = 7, |22 − 60| = 38, and |23 − 60| = 37. Now the density is the same as some reflex-angled triangles which share the same great circles, respectively: (2/3 3 3) or (2 3 3/4); (2/3 3 5/2), (2 3/4 5/3) or (2 3 5/7); (3 3 5/8) or (3 3/4 5/3).

It's interesting that these cases are exactly the ones where strict application of the Wythoff symbol gives strange results for the vertex figure. I think that means they have to be corrected based on these considerations, using the Schwarz triangles that actually have the same density. In particular, Coxeter writes that for quitco and quitdid, all the faces are retrograde save the aforementioned hexagons and decagons; to my mind that implies true vertex configurations of 4/3.6.8/5 and 4/3.10.10/7. As for the reflex-angled cases, first we'd have to define what Schläfli symbols {n} with n ≤ 1 mean; hopefully they then produce the geometrically correct results! Double sharp (talk) 15:41, 16 March 2015 (UTC)

P.S. This is presumably why Stella gives quitco and quitdid negative density values (see above). Double sharp (talk) 16:18, 27 July 2019 (UTC)