Talk:Runcinated pentachoron
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The symbols given for the cells -- {3,3} and 3,4,4 -- are confusing because they belong to different systems. I've (privately) used "{3}×{}" for the prism, i.e. the Cartesian product of a triangle {3} and a line segment {}. --Anton Sherwood 03:46, 3 January 2006 (UTC)
- {3,3} and 3.3.3 are different notations for the same system - defining the vertex figure as a sequence of faces around a vertex. {p,q} is just a short-hand for "regular" cases when a vertex has all the same faces. {p,q} = p^q = p.p.p....p (q times). For example {3,5}=3^5=3.3.3.3.3=icosahedron. And for nonregular uniform polyhedra like icosidodecahedron=3.5.3.5 for alternating triangles and pentagons on a vertex.
- I've found this notation very use for the uniform polyhedra and vital for building the article List_of_Uniform_Polyhedra with nonconvex uniform polyhedra which it's much harder to see the vertex figures.
- The same notation works for polychora "edge figures" as well (Sequences of cells around an edge). {p,q,r}={p,q}.{p,q}.{p,q}... (r times) {3,3,5} means 5 tetrahedra per edge. Unfortunately it's less useful for uniform polychora in general because there can be more than one type of edge, while uniform polyhedra have only a single vertex type.
- I agree it is less than clearly explained now and deserves some explanation OR using the same (full) notation for all. Writing 3.3.3.3.3 instead of shorthand {3,5} for instance. Of course you can argue writing out icosahedron directly is even more clear. I considered that but figured rolling over the notated text does show the link to the name as well, and some polyhedra have LONG names!
[edit] a detail about runcination
A runcinated {p,q,r} consists of {p,q}, p-prisms, r-prisms and {r,q}, cells of the dual. When the vertex figure is {3,3} the dual cell is also {3,3}, but we want to avoid giving the idea that the vertex figure appears as a cell in the general case. —Tamfang 06:54, 23 February 2006 (UTC)
