Talk:Vertex figure

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[edit] 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)

[edit] 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)

[edit] 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)