Schläfli symbol

Schläfli symbols are a notation used to represent regular polytopes. The succinctly describe regular polytopes including regular Euclidean tilings and regular hyperbolic tilinga, as well as the more ordinary spherical polytopes. Various extensions to Schläfli symbols exist to represent wider arrays of polytopes.

Description
A Schläfli symbol consists of several numbers in sequence (usually separated by commas) enclosed within curly brackets. The numbers can be positive integers, fully reduced positive fractions, or infinity. The simplest Schläfli symbols just use positive integers. These simple Schläfli symbols can be defined recursively. As a base case, $$\{\}$$ is the dyad, and $$\{n\}$$ is a regular $n$-gon. Then for rank 3 and higher the a symbol $$\{p_1, p_2, p_3, dots, p_{n-1}, p_n\}$$ is a tiling of polytopes with the symbol $$\{p_1, p_2, p_3, dots , p_{n-1}\}$$ with $$p_n$$ of them placed around each $$n-2$$ element. For example, the dodecahedron is $$\{5,3\}$$. That means its faces are $$\{5\}$$, i.e. pentagons, and there are $3$ of them around each vertex. $$\{4,3,4\}$$ describes a regular polychoron with 4 cubes around each edge, i.e. a cubic honeycomb.

Although useful for picturing things, instead of counting the number of facets around elements this can be done in terms of vertex figures. As before $$\{\}$$ is the dyad, and $$\{n\}$$ is a regular $n$-gon, however additionally $$\left\{\frac n m\right\}$$ is a star polygon with $n$ vertices with each vertex connected by an edge to the vertices $m$ steps away. Note that $$\left\{\frac n 1\right\}$$ is exactly equivalent to $$\{n\}$$. Then for higher dimensions the symbol $$\{p_1,p_2,p_3,\dots,p_n\}$$ represents the polytope with faces of $$\{p_1\}$$ and a vertex figure of $$\{p_2,p_3,\dots,p_n\}$$. That is the first value gives the face of the polytope and the rest of the values are the vertex figure. This allows for polytopes like the great dodecahedron which has the symbol $$\left\{5,\frac 5 2\right\}$$.

Conversion to Coxeter-Dynkin diagrams
Schläfli symbols can be easily converted to a Coxeter-Dynkin diagram. A regular polytope has linear Coxeter-Dynkin diagram with the only the first node ringed, so to convert a Schläfli symbol to a Coxeter-Dynkin diagram you create a linear diagram with the first node ringed and then label the edges in between with the values of the Schläfli symbol in order. The Schläfli symbol $$\{n,m,\dots,z\}$$ becomes the diagram .... This includes cases where the Schläfli symbol has fractional or infinite values.

Extended Schläfli symbols
Coxeter later adapted the Schläfli symbol to quasiregular polytopes as well: Again a single ringed node would occur in its Coxeter-Dynkin diagram. Then the diagram gets folded and displayed such that its single ringed node is displayed to the very left, while the two legs would run to the right. The according Schläfli symbol is then a left curly bracket followed by the two left-aligned number sequences of those legs, one atop the other, and finally closed by the right curly bracket: $$\left\{\begin{aligned}&P, Q, R \\&S, T\end{aligned}\right\}$$ = oRoQoPxSoTo. He further even allowed thereby for bifurcation nodes in the diagram (as being used for the Gosset polytopes). Then any of the left-aligned number sequences might split up into two (or more) lines from some point on.

Sure, none of these extended usages is better usable within inline texts as the according graphical devices of the Coxeter-Dynkin diagrams. Moreover they always are restricted to a single ringed node, while the latter are much more general. This is why the extended Schläfli symbols mostly became abandoned in favor of the latter.