Talk:Vertex configuration

undated old comment

i am new to wikipedia, but i allowed myself to erase the following sentence: "So 3.3.4.3.4 represents the snub cube starting with 2 triangles." this is clearly wrong. the snub cube vertex configuration is 3.3.3.3.4.

Right, 3.3.4.3.4 is the snub square tiling. —Tamfang (talk) 01:53, 18 February 2018 (UTC)

Vertex symbol?

This term Vertex configuration is not standardized, and I've been hunting for better alternatives, best so far vertex symbol. It also could be considered a Schläfli symbol which is normally written as {p,q} for regular polyhedra, which could as well be written as p^q in this notation. I don't have any original write up for Schläfli symbol origin, mainly uses by Coxeter for both notations.

Web search:

[1] Vertex Symbol - Each regular or semi-regular polyhedron has identical sequences of regular polygons surrounding each vertex. This sequence identifies the polyhedron, and can be used to assign a vertex symbol to it. The symbol is a sequence of numbers that shows the sequence of polygons about each vertex. Each number represents a face that is a regular polygon with that many sides.

[2] Schläfli Symbol - The Schläfli symbol encodes the topology of a 2D tiling, or equivalently, a 2D periodic network. A Schläfli symbol is given for each symmetrically distinct vertex and lists the number of edges of each tile (the "ring size"), for all tiles coincident with that vertex. The ring sizes are given in cyclic fashion around the vertex so that adjacent entries refer to tiles with a common edge. Thus, (6.6.6) refers to the hexagonal tiling of the 2D euclidean plane, (5.5.5) to the edge skeleton of the dodecahedron (which is a tiling of the sphere).

[3] Schläfli symbol A notation, devised by Ludwig Schläfli, which describes the number of edges of each polygon meeting at a vertex of a regular or semi-regular tiling or solid. For a Platonic solid, it is written {p, q}, where p is the number of sides each face has, and q is the number of faces that touch at each vertex. (Implies semiregulars can also be written, even if no example given.)

[4] & [5] - Vertex description - A way of describing a uniform polyhedron, by giving the sequence of face types meeting around a vertex. Eg "4, 4, 4" represents a cube, because there is a sequence of three regular 4-sided polygons (squares) around each vertex. Most people prefer to use dots rather than commas to separate the faces, but some reason I like commas a lot more. Each face is specified as n/p, where n is the number of sides, and p is the number of times the polygon encloses its centre. Eg 5/2 means a pentagram. When p is 1 we leave it out. To specify a retrograde face, use n/(n-p), eg 5/3 for a pentagram which goes the opposite way around the vertex. Finally, if the vertex is surrounded more than once by the faces, put it in parenthases and add /q at the end, where q is the number of times the faces circle the vertex.

There's a similar issue for face-transitive dual polyhedra/tilings with face configuration.

Thoughts? Tom Ruen 00:45, 11 May 2007 (UTC)

"Vertex symbol" is OK, "Vertex description" would be my favourite. "Schläfli symbol" is not appropriate: one of the most important characteristics of Schläfli symbols is their duality - swap it back to front and you have the symbol for the dual polyhedron. Vertex descriptions do not do this. The first bit of web info you trawled is quite wrong: the form (a.a.a) is not a Schläfli symbol. Coxeter produced what he called "modified Schläfli symbols" for some quasiregular solids, but even this notation cannot be used for all the quasiregulars, never mind the semiregulars. The snippet on Vertex Descriptions gets the last bit wrong (though since it's Rob Webb, he may make his variation accepted by default). It harks back to the notation used by Cundy & Rollett, where they used say (3.4)² for the cuboctahedron, or 4.3³ for the snub cube. This also does not generalise comfortably to all the uniform stars. Nowadays we write 3.4.3.4 or (3.4.3.4), and 4.3.3.3 or (4.3.3.3). Re face-transitives, i.e. uniform duals, I have seen the same symbols in square brackets, eg [3.4.3.4] and [4.3.3.3]. HTH -- Steelpillow 20:06, 11 May 2007 (UTC)

"Similarly when four faces meet at each vertex, p.q.r.s, if one number is odd the rest must be equal."

This is clearly wrong since 3.4.5.4 exists, but there's obviously _some_ topological constraint on four faces... Someone else already put an inline comment in the page, but it's been sitting there unnoticed, so I figure I should mention it on the talk page too. 24.61.207.84 (talk) 12:52, 24 January 2016 (UTC)

An odd face cannot have dissimilar neighbors? —Tamfang (talk) 01:49, 18 February 2018 (UTC)

That's not right either; see snub cube for example. —Tamfang (talk) 23:04, 18 July 2023 (UTC)