Schlafli symbol determining number of faces

Question

Schlafli symbols are used to describe polytopes, but can also be used to describe more general objects through the use of flags. In particular, some information can be readily 'read-off' from a Schlafli symbol (like the number of edges in a face).

I was wondering how the number of faces in a volume is encoded in a Schlafli symbol. For example, we know that

a tetrahedron $\{3,3\}$ has $4$ faces,

an octahedron $\{3,4\}$ has $8$ faces,

a cube $\{4,3\}$ has $6$ faces,

and an icosahedron $\{3,5\}$ has $20$ faces.

Without considering the interior angles, is there a way to read off this face information from the Schlafli symbol by considering flag orbits?

Thanks!

Answer 1

This has nothing to do with flag orbits, but if you just want to count faces (and, while we're at it, vertices and edges) ...

Let a $\{p,q\}$-hedron have $f$ faces (each a regular $p$-gon), $v$ vertices (each of degree $q$), and $e$ edges. Then $pf$ counts each edge twice, as does $qv$: $$pf = 2e= qv \tag{1}$$ Moreover, Euler's polyhedron formula tells us $$v-e+f=2 \tag{2}$$ So, we have three equations in three unknowns, which we can readily solve:

$$v = \frac{4 p}{2(p+q) - p q} \qquad e = \frac{2 p q}{2(p+q)-p q} \qquad f = \frac{4 q}{2(p+q) - p q} \tag{$\star$}$$

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As for gleaning more information from Schläfli symbols, this question mentions a formula for the dihedral angle of a $\{p,q\}$-hedron: $$\sin\frac{\theta}{2} = \frac{\cos(\pi/q)}{\sin(\pi/p)}$$ My answer gives the extension for the dichoral angle of a $\{p,q,r\}$-tope.

Answer 2

There even is a generalization of Schläfli symbols (for regular polytopes) towards Coxeter-Dynkin diagrams for the whole set of Wythoff constructable polytopes. Wythoff provided a kaleidoscopical construction advise, which allows to describe all Stott expanded versions of the regular polytopes as well. This encompasses nearly all uniform polytopes (only snubs and some further odd ones are out of scope here).

Again you could ask the same question wrt. to those numbers of the Coxeter-Dynkin diagrams, deriving therefrom the number of vertices (surely just a single type), the number of edges (of each symmetry inequivalent type), the number of polygonal faces (of each type), etc.

This is where my Website https://bendwavy.org/klitzing/home.htm provides lots of according infos. E.g. consider https://bendwavy.org/klitzing/explain/dynkin.htm for the General setup of those Coxeter-Dynkin diagrams. Or https://bendwavy.org/klitzing/explain/incmat.htm on the subject of incidence matrices, which are providing nothing but the more detailed infos of your above set v, e, f. Details on the dihedral angles (as mentioned by @Blue) you could find at https://bendwavy.org/klitzing/explain/dihedral.htm.

The remainder of this website, beneath further explanatory pages, then is devoted to lots of individual pages for various individual polytopes... Probably a huge fundus to dive in.

--- rk