Talk:Small stellated dodecahedron

2009 comments

How many Points (and faces) are in this polyhedron?
--Lyamk (talk) 04:47, 7 February 2009 (UTC)[reply]

It's given in the table: Faces = 12, edges = 30, vertices = 12. Tom Ruen (talk) 03:35, 8 February 2009 (UTC)[reply]

How can this be a spherical tiling? The sphere has Euler characteristic 2 while the small stellated dodecahedron has Euler characteristic -6, so how can it be a spherical tiling? Double sharp (talk) 06:38, 14 August 2009 (UTC)[reply]

See: Kepler-Poinsot_polyhedra#Euler_characteristic_.CF.87 Tom Ruen (talk) 07:17, 14 August 2009 (UTC)[reply]

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Confusion about truncated small stellated dodecahedron

Discussion of the "truncated small stellated dodecahedron" is confusing ... The picture is of a normal dodecahedron, and neither the text nor the picture seem to apply to the polyhedron at [1]. Seems to require revision, if a proper figure can be made. — Preceding unsigned comment added by Nernom (talk • contribs) 19:21, 22 January 2018 (UTC)[reply]

It is confusing. A small stellated truncated dodecahedron, t{5/3,5},

is different than a "truncated small stellated dodecahedron", t{5/2,5},

. The second looks like a regular dodecahedron, except the pentagon faces are doubly-wrapped pentagons, {10/2}. To see this you'd have to show progressively larger truncations of a pentagram until pairs of vertices and edges converge. Tom Ruen (talk) 20:13, 22 January 2018 (UTC)[reply]

Grünbaum has a nice picture of the truncation procedure applied to {5/2, 5} in his famous paper; I think we can use it as an illustration. For this reasons the small stellated truncated dodecahedron is sometimes called the quasitruncated small stellated dodecahedron, as it is t{5/3, 5} (the cutting planes have passed right through the centre and are now overlapping each other coming back out the other way). Double sharp (talk) 00:06, 23 January 2018 (UTC)[reply]

I edited the description a bit, although it's a bit wordy now. See if you think it helps to clarify. RobertCWebb (talk) 01:55, 23 January 2018 (UTC)[reply]

Yes, thanks! The additional information is appreciated ... there are still things I'm confused about though, which I think any non-geometer would be confused about: (1) it looks like the previous paragraph, "This polyhedron is the truncation of the great dodecahedron:" applies to the TSSD; does it? That should be clarified. (2) what does it mean that it "looks like a regular dodecahedron on the surface"? If its surface is (looks?) identical, what is the point of making a distinction? If the regular dodecahedron figure, present now, is correct, I argue the TSSD should be removed from the article. If the regular dodecahedron figure is incorrect, the picture should be corrected. In an intermediate case that e.g. the difference between the regular dodec and the TSSD is that the TSSD has additional faces that do not change the solid object, I argue that the TSSD should be removed. Nernom (talk • contribs)

Thanks Rob. I think its good now. Can you make a nonuniform truncation image that shows the geometry when its not degenerate? Tom Ruen (talk) 18:25, 23 January 2018 (UTC)[reply]

Yes, much improved! But agreed, a non-degenerate figure would be great -- necessary, I'd argue; a solid with exactly the same dimensions as a dodecahedron seems too esoteric to be included. One can always complicate a given solid by adding overlapping faces. Nernom (talk • contribs)

I think it would be a serious mistake to remove t{5/2, 5} just because it is degenerate as a uniform polyhedron. Granted, one can make an overlapping-face version of any polyhedron, but this one turns up naturally in the truncation sequence of a regular polyhedron, and not including it frankly raises more questions than it would answer. Anyway, t{5/2, 5} looks exactly like a dodecahedron, but has two faces for every one of the dodecahedron; one pentagon {5} (from the vertices of {5/2, 5}) and one decagon {10/2} (from the faces of {5/2, 5}). Since the latter looks exactly like two coinciding pentagons the resulting solid looks like 3 coinciding dodecahedra, but the way the faces connect to each other is of course quite different. (About the same thing happens for t{5/2, 3}). Double sharp (talk) 23:47, 23 January 2018 (UTC)[reply]

BTW, since quite a few people seem to stumble over this (so did I at first, though that was back in 2009), it occurs to me that we should show a complete series through the hyper-, quasi-, and antitruncations for {5/2, 5} just as we do for {4, 3} (the cube) at quasitruncation, and that {5/2, 3} should get the same treatment. All this seems to be because the small stellated truncated dodecahedron t{5/3, 5} is just rather poorly named. Double sharp (talk) 23:52, 23 January 2018 (UTC)[reply]

Really all of the star polyhedra may seem too esoteric given I overlooked them for 15 years, not understanding what they were, since overlapping geometry is hard to understand, while its the underlying topology that is what defines them rather than appearance. DS, I'm open for more images of varied truncations if someone wants to generate them. Tom Ruen (talk) 00:01, 24 January 2018 (UTC)[reply]

Small stellated dodecahedron to great dodecahedron truncation sequence	Great stellated dodecahedron to great icosahedron truncation sequence

@Tomruen: My copy of Stella died with my last computer and I'm not sure how to generate all those extra truncations with it anyway, but certainly the truncation series from {5/2, 5} down to t{5/2, 5} is doable with its dual-morph feature. I could upload Grünbaum's illustration for that, too. I find it's not that difficult to think about this degeneracy once you realise that the truncation literally cuts off the spikes and leaves you with what looks like the original dodecahedral core, but intermediate steps would certainly help. There is, I think, more challenge in illustrating where t{5/3, 5} comes into this; do you have an idea on how to do that? Double sharp (talk) 02:21, 24 January 2018 (UTC)[reply]

I never used the truncation morph before. Tom Ruen (talk) 03:08, 24 January 2018 (UTC)[reply]

@Tomruen: Yes, that's excellent! Would you mind also doing it for the great stellated dodecahedron, which passes through a similar degenerate figure along the way as well? Double sharp (talk) 04:10, 24 January 2018 (UTC)[reply]

Added. This is what Stella generates. I suppose it would be better as a loop, or at least a pause at start and end. Tom Ruen (talk) 04:22, 24 January 2018 (UTC)[reply]

@Tomruen: A loop would be welcome, but I think it's not necessary if it's difficult to generate. I would nevertheless prefer it if the gif went more slowly; around the endpoints it is okay, but around the rectification in the centre it really moves too fast for the eye to follow, even if you already know what you're looking for. Double sharp (talk) 04:49, 24 January 2018 (UTC)[reply]

Okay, I've added Tom's gifs to all the relevant polyhedra in the truncation sequences. Now the more vexing question is how to illustrate where the quasitruncations t{5/3, 5} and t{5/3, 3} come into this (not to mention the appearance of the quasicantellates and the quasicantitruncates further in this series – the names "quasirhombicosidodecahedron" and "quasitruncated dodecadodecahedron" surely make it clear that this is not just OR). As a first step I would note that the original pentagram faces get quasitruncated exactly as they would in isolation (I think you have a gallery of those polygram morphs), while the faces of the dual expand to connect them, but visualising that is much easier said than done. (It goes without saying that this applies just as well to the nonconvex uniform polyhedra arising from the cube.) Double sharp (talk) 05:27, 24 January 2018 (UTC)[reply]

I think I'm done for now, a good addition. I'm sure I could do more with some extended truncations as calculated forms, but nothing quick. Tom Ruen (talk) 05:42, 24 January 2018 (UTC)[reply]

zero-based indexing?

It is the second of four stellations of the dodecahedron.

Funny, the linked table says it's the first of three. —Tamfang (talk) 03:05, 4 December 2019 (UTC)[reply]

It honestly seems more natural to me to say that the dodecahedron is the zeroth stellation of itself. Then {5/2, 5} is the first, {5, 5/2} is the second, and {5/2, 3} is the third. However, it's not terribly obvious how to number the stellations of a more general polyhedron (the regular icosahedron will do as an example) this way, as then the stellation diagram is not just a straight line. I'd prefer "first" to make it clear that it's the first in the very same sense as that of polygons; it's what you get when you extend the edges of the dodecahedron until they meet again, just like how you construct a pentagram. Double sharp (talk) 16:46, 4 December 2019 (UTC)[reply]