Talk:Vertex figure

4-polytopes and 3-honeycombs

I made a vertex figure symmary chart on a talk subpage for reference:

Talk:Vertex_figure/polychoron

Tom, has any of this information been published elsewhere? Dinogeorge's website has disappeared into web.archive.org and Johnson's book is not yet published, so neither is a suitable reference - even if they give the vertex figures, which I doubt. Jonathan Bowers' site gives labelled perspective images, but no other info. Your page is all very nice but as far as I know it is original research and Wikipedia is the wrong place for it. -- Cheers, Steelpillow (Talk) 20:14, 27 August 2009 (UTC)

Hi Guy! I'm in luck this time for a published source! I copied the idea from Symmetries of Things, 2009, which drew these vertex figures for the uniform 4-polytopes, AND Euclidean honecombs. Tom Ruen (talk) 20:34, 27 August 2009 (UTC)

Dorman Luke construction

Having just added a picture and some explanation, it occurs to me that it is also relevant to dual polyhedra. Should it:

1. Stay here?
2. Move to Dual polyhedron?
3. Move to its own brand new page?

-- Steelpillow 20:15, 3 June 2007 (UTC)

Nice picture. I was thinking also better fit for Dual polyhedron. Tom Ruen 23:50, 3 June 2007 (UTC)

Done. -- Steelpillow 21:06, 4 June 2007 (UTC)

Imaginary?

To quote a current snippet:

By considering the connectivity of these neighboring vertices, a full imaginary (n-1)-polytope can be constructed for each vertex of a polytope:

* Each vertex of the vertex figure coincides with a vertex of the original polytope.

* Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two alternate vertices from an original face.

* Each face of the vertex figure exists on or inside a cell of the original n-polytope (for n>3).

* ...and so on to higher order elements in higher order polytopes.

What is "imaginary" supposed to mean in this context? Perhaps "skew" is meant. And what does the construction described have to do with vertex figures? It seems quite trivial and has no real theoretical or practical significance that I am aware of. I think this should not be here, and I will delete it in a few days if nobody comes to its defence (or deletes it first). -- Steelpillow 18:50, 22 May 2007 (UTC)

By imaginary, I meant it wasn't a direct element of the polytope. A cube is made of squares, but has an imaginary triangle vertex figure since there's no faces that are triangles. Tom Ruen 02:03, 23 May 2007 (UTC)

Well, "imaginary" is th wrong word - in mathematics it implies something to do with the square root of minus one, which is not the case here. Further, once constructed the polytope is no longer "imaginary". I think the word can just go. As for the rest, I misunderstood it first time round - it's correct. -- Steelpillow 17:03, 23 May 2007 (UTC)

Edge figures

Another quote:

Edge figures

In higher order polytopes other lower order figures can be useful. For instance an edge figure of a polychoron or 4-honeycomb is a polygon representing the set of faces around an edge. For example the edge figure for a regular cubic honeycomb {4,3,4} is a square, and for a regular polychoron {p,q,r} is the polygon {r}.

Is this name "edge figure", correct? Can anybody provide a reference? It's important not to delete this un-referenced material without checking, because the (2D) dual of this figure is a face of the dual polychoron or honeycomb (e.g. by Dorman Luke's construction). But has it got the correct heading? -- Steelpillow 19:06, 22 May 2007 (UTC)

I don't have a clear defined reference myself, but took it as an extrapolation of "vertex figure". A polyhedron is created by wrapping a closed sequence of faces around every vertex (creating vertex figures). A polychoron is created by wrapping a closed sequence of cells around every edge (creating edge figures). So a vertex figure is defined by the directional set of edges around every vertex and an edge figure is defined by the directional set of faces around an edge. That's my interpretation anyway. Tom Ruen 02:00, 23 May 2007 (UTC)

That's my understanding too. Some questions: Does this term exist in the literature? If not, should we be using it. And if we should, dos it deserve a separate entry in its own right? -- Steelpillow 17:03, 23 May 2007 (UTC)

I put it as as section here for that reason of limited information along with strong connection to vertex figures. Its used in articles like List_of_regular_polytopes#Five-dimensional_regular_polytopes, along with face figures too! Tom Ruen 18:59, 23 May 2007 (UTC)

3d example

I think for purposes of this article a solid picture illustrates the concept better than a Schlegel diagram. —Tamfang (talk) 18:27, 25 December 2008 (UTC)

Edge figures

I'm not sure that edge figures are adequately represented in reliable sources. In particular Dr. Klitzing does not appear to have sufficient record of peer-reviewed publications for his web site to be an authoritative source. The fact that a few of us talk about them informally is also not sufficient. As one illustration of the reasons behind this attitude consider that the vertex figure of a higher-dimensional edge figure is not so much to do with an edge as with an (n-2)-cell; in this context the term "edge figure" is inappropriate and its usage immature. I should like to see either a clear path towards adequate referencing, or deletion of this section. — Cheers, Steelpillow (Talk) 18:46, 24 May 2011 (UTC)

I moved as a section from Edge figure, still looking for source support/usages. It's linked now on a number of articles, mostly higher honeycombs, and List_of_regular_polytopes. Tom Ruen (talk) 19:42, 24 May 2011 (UTC)

Hi Tom. I'm not convinced that cross-linking represents a move towards verifiable sources. If the phrase is useful, by all means explain it on the page it is being used on, but if it's not out there in the literature we can't pretend that it is encyclopedic. — Cheers, Steelpillow (Talk) 20:00, 25 May 2011 (UTC)

Two sources using the term "edge figure" are Algebraic decomposition of non-convex polyhedra, a conference paper from 1995, and Facets and Rank of Integer Polyhedra a paper in the book M. Jünger, G. Reinelt (eds.), Facets of Combinatorial Optimization, DOI 10.1007/978-3-642-38189-8_2, © Springer-Verlag Berlin Heidelberg 2013. But I haven't parsed them to to point of verifying that the edge figure constructions in those sources are the same as this article. But overall, the term seems quite rare. --Mark viking (talk) 19:33, 5 April 2015 (UTC)

single-ringed

What is single-ringed? At least, where I may read about it? Is it about mirrors in Coxeter diagrams?Jumpow (talk) 12:37, 3 November 2015 (UTC)

Yes Tom Ruen (talk) 15:20, 3 November 2015 (UTC)

Also not fully clear next sentence:

Regular and single-ringed uniform polytopes will have a single edge figure type, while in general, a uniform polytope can have as many edges as active mirrors in the construction...

Edges in second case means figure type?Jumpow (talk) 14:34, 3 November 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Vertex figure/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This page needs a lot of work - to add a lot of important elementary information, and to tidy-up and expand the more advanced stuff already here. -- Steelpillow 18:52, 22 May 2007 (UTC)

Last edited at 18:52, 22 May 2007 (UTC). Substituted at 20:19, 1 May 2016 (UTC)

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Klitzing

Is Richard Klitzing's web site a WP:RS for the "edge figure"? The article currently cites the page at http://www.bendwavy.org/klitzing/explain/verf.htm — Cheers, Steelpillow (Talk) 18:14, 17 March 2018 (UTC)

Images of solids

Icosidodecahedron (light) and its dual (dark). The base of each rectangular pyramid is its vertex figure.

Currently this article does not contain a single image of a polyhedron to illustrate its vertex figure. I tried to change this by including three images of Archimedean solids and their duals, including the one on the right. This was reverted by Steelpillow. I think dual compounds are a good example, and cutting in the middle of edges is one of the methods described in the article. Any better ideas? Watchduck (quack) 18:38, 17 March 2018 (UTC)

Compounds are not vertex figures, they do not illustrate the basic idea. They need a complicated caption like the one you give here, and even then you have still not explained it clearly. If you actually want to illustrate the relationship to duality, check out the diagram in the article on the Dorman Luke construction. There are a fair number of illustrations of polyhedra on the commons showing vertex figures, which would be more appropriate here. But I think they are badly designed as introductory diagrams so I am preparing some which I hope will be more suitable for this article. — Cheers, Steelpillow (Talk) 19:44, 17 March 2018 (UTC)

Four are now in the article. I prefer basic graphics because colours can be distracting depending on the context. Do they look any use? — Cheers, Steelpillow (Talk) 19:58, 17 March 2018 (UTC)

Looks good to me. Thanks! Watchduck (quack) 21:46, 17 March 2018 (UTC)

Illustrations

It would be interesting to hear your opinion on how the vertex figures of Archimedean solids should be illustrated in their respective articles: Talk:Archimedean solid#Images Greetings, Watchduck (quack) 02:03, 16 April 2018 (UTC)

They are usually illustrated as plane polygon diagrams, with no attempt to place them in perspective on the polyhedron (once the basic principle has been explained). Wikipedia seems to be doing the same, including them in the infoboxes and linking to this article for its explanation of the basic principle. So that aspect is OK. But gratuitous colouring, shading or outlining of the associated faces makes them look like pyramids seen from above and that is quite wrong. Once again, a simple outline drawing with clear labelling is the time-honoured and far the best solution. In my very firm opinion, they all need redrawing without the "bells and whistles" distractions. — Cheers, Steelpillow (Talk) 08:36, 16 April 2018 (UTC)

Current illustration at Great icosahedron

Plain line drawing

Annotated line drawing

Cuboctahedron showing a vertex figure

By way of illustration: [first 3 images on the right]

I submit that the line drawings are clearer and more useful. The last one, with annotations, is really only suitable for polyhedra with regular faces. I am not sure whether the infoboxes are better standardised on plain drawings or annotated where it makes sense. — Cheers, Steelpillow (Talk) 09:01, 16 April 2018 (UTC)

@Tomruen: Didn't you draw most of the current ones? — Cheers, Steelpillow (Talk) 09:11, 16 April 2018 (UTC)

But if you still want to see a perspective view of the vertex figure in place, then I'd suggest a line drawing of the half-edge type, something like this: [rightmost image]

— Cheers, Steelpillow (Talk) 09:30, 16 April 2018 (UTC)

(I took the liberty to format your images in a more compact way.) This kind of image is good to explain what a vertex figure is. But we should be able to agree that the vertex figure of a particular solid should be shown in a way that correctly represents its side lengths and angles. In a context where the full solid is also shown, I find the representation as pyramid base quite ok. I am just talking about convex solids here btw — actually just [mainly] about Archimedean solids. Watchduck (quack) 12:49, 16 April 2018 (UTC)

Absolutely not. The vertex figure does not include the vertex itself and it includes the connected edges only as corners (you are perhaps thinking of a vertex star). Nor do I think that confining yourself to thirteen specific polyhedra is likely to yield a generally acceptable solution. — Cheers, Steelpillow (Talk) 13:27, 16 April 2018 (UTC)

Right. That restriction was not really necessary. What matters to me is that the image of the vertex figure is shown in context with an image of the full solid. That's why I think that consistency of face colors is a good idea. The vertex figure is usually described as the result of cutting off a corner of the polyhedron. So it seems legit to me to represent it by showing the part that was cut of. I don't think this creates the impression that the vertex figure is actually a pyramid. Again: This is not about explaining what a vertex figure is, but about showing the vertex figure of a particular solid. So we can expect the reader to already know what a vertex figure is. Watchduck (quack) 14:23, 16 April 2018 (UTC)

I am sorry, that strikes me as utterly flawed. On the one hand you just want to illustrate the vertex figure. On the other you are proposing to illustrate a pyramid and pass it off as a vertex figure on the assumption that readers will see through the sleight of hand. That is not illustrating a vertex figure, that is madness. — Cheers, Steelpillow (Talk) 15:56, 16 April 2018 (UTC)

I suppose images like those with the black polygons on the right would seem better to you (although you probably still find them excessively gaudy), because they are closer to the literal definition. But although they might be better to unambiguously show what a vertex figure is as a concept, I don't think they are better to show what the vertex figure of a particular solid is.

No, I am not proposing to illustrate a pyramid. I am proposing to show a pyramid to illustrate its base. That does not seem crazy to me. An orthogonal projection of a pyramid from above contains its base, and is therefore an illustration of it. Whether or not it is a good illustration depends on the question if the triangles are just visual noise, or if they convey useful additional information. In this context I think they do. But I got your point. Lets see if there are other opinions. Watchduck (quack) 17:16, 16 April 2018 (UTC)

with labels

PS: I would like to mention that Figure 21.1 (p. 289) in The Symmetries of Things by Conway et al. illustrates vertex figures in exactly the way I have done here — by showing a pyramid cut from the solid. Watchduck (quack) 23:24, 26 July 2018 (UTC)

TSoT also shows vertex figures with number labels for each p-gon. Tom Ruen (talk) 00:40, 27 July 2018 (UTC)

I am not opposed to including labels. How about something like the image on the right? (Shown as 140px like in the infobox.) Watchduck (quack) 16:35, 27 July 2018 (UTC)

@Watchduck: You are cherry-picking snippets to try and back your case. The text in TSOT accompanying the illustrations on p.289 states categorically that "the vertex figure of the cuboctahedron (Figure 21.1 left) and icosidodecahedron are rectangles whose sides are numbered ...". The drawing shows a slice through a polyhedron, the "pyramid" is merely that part to one side of the slice, with the overall polyhedral form remaining intact. It does not, as you claim, merely "show a pyramid cut from the solid". Note the close correspondence between Fig 21.1 and the cuboctahedron drawing I mentioned above.

I am not sure if polychora have been mentioned here, but if a pyramid does occur as a vertex figure then it is of some four-dimensional polychoron. It is, as I have said, mathematical madness to use the same pyramid in an attempt to illustrate the lower-dimensional vertex figure of a polyhedron. The central point adds nothing to the description here, save confusion.

— Cheers, Steelpillow (Talk) 17:06, 27 July 2018 (UTC)

I really don't intend to reopen the discussion. I just saw these images by coincidence, and added them here as context.

Anyway: "The drawing shows a slice through a polyhedron, the 'pyramid' is merely that part to one side of the slice" is exactly how I would describe my own images. (The rest of the polyhedron can be seen through the translucent layer. Except for triangles the base does not have cylinder edges.) I don't think they are similar to images like this — while you do. Nothing more to say.

My question to Tomruen is only about the Archimedean solid infoboxes, and not about the illustration of vertex figures in general. Watchduck (quack) 17:32, 27 July 2018 (UTC)

You put up one image to canvass opinion and then say you are discussing a different one. You ask "my question to Tomruen" as if the rest of Wikipedia is expected to let you boys share ownership. You move the discussion here from Talk:Archimedean solid#Images and then wonder why it becomes a discussion on the present topic. The only constancy in all this nonsense is the single-minded pursuit of an untenable position. No. — Cheers, Steelpillow (Talk) 18:04, 27 July 2018 (UTC)

Goodness. My question to Tomruen is here, because his comment was here. I now copy both to Talk:Archimedean solid#Images, so we can talk there about that specific topic. No ownership implied or wanted. The discussion between us two (Steelpillow and Watchduck) has long reached a dead end. We both made our points, and there is nothing more we can do. Watchduck (quack) 18:46, 27 July 2018 (UTC)

Does the vertex figure include the vertex itself?

Some parts of the article imply Yes, and some parts imply No.