Talk:Vertex figure

message: 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  is the wrong place for it. -- Cheers, Steelpillow (Talk) 20:14, 27 August 2009 (UTC)
replies: text: 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)
thread_title:  4-polytopes and 3-honeycombs 

message: Having just added a picture and some explanation, it occurs to me that it is also relevant to dual polyhedra. Should it:
#Stay here?
#Move to Dual polyhedron?
#Move to its own brand new page?
-- Steelpillow 20:15, 3 June 2007 (UTC)
replies: text: Nice picture. I was thinking also better fit for Dual polyhedron. Tom Ruen 23:50, 3 June 2007 (UTC)
replies: text: Done. -- Steelpillow 21:06, 4 June 2007 (UTC)
thread_title: Dorman Luke construction

message: 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)
replies: text: 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)
replies: text: 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)
thread_title: Imaginary?

message: 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)
replies: text: 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)
replies: text: 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)
replies: text:  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)
thread_title: Edge figures

message: 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)
thread_title:  3d example 

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)
thread_title:  Edge figures 

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

text: 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)
thread_title:  single-ringed 

message: Substituted at 20:19, 1 May 2016 (UTC)
thread_title: Assessment comment

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thread_title:  External links modified 

message: 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)
thread_title:  Klitzing 

message: 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)
replies: text: 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)

text: 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)
replies: text: Looks good to me. Thanks! Watchduck (quack) 21:46, 17 March 2018 (UTC)
thread_title: Images of solids