Schläfli symbol

Schläfli symbols are a notation used to represent regular polytopes. The succinctly describe regular polytopes including regular Euclidean tilings and regular hyperbolic tilings, 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
Many extensions to the basic Schläfli symbols exist. These extensions allow symbols to represent more regular polytopes or other classes of polytope altogether.

Coxeter's extension
Coxeter extended Schläfli symbols to represent quasiregular polytopes as well. Coxeter's extension allows Schläfli symbols to have up to two rows of values between the curly brackets. A single row still represents the same polytope. If the symbol has two rows it is converted to a Coxeter-Dynkin diagram with a single ringed node. The ringed node is placed on the right and two linear branches extend off to the left. Each value in the symbol is placed between two nodes, with values from the top row being placed on the top branch in order and values from the bottom row being placed on the bottom branch in order.

$$\left\{\begin{aligned}&p, r, t \\&q, s\end{aligned}\right\}$$ = =

Coxeter 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.

Wythoffian prefixes
Schläfli symbols have also been extended to represent Wythoffian polytopes using prefixes representing Wythoffian operations. In general any Wythoffian operation is assigned the prefix $t$ with some number of subscripts representing the indices of ringed nodes in the Coxeter-Dynkin diagram. Indices begin at 0 so $$t_0$$ rings the first node.

For example the cuboctahedron (CDD: ) has the symmetry group. So we start with the symbol $$\{4,3\}$$ (CDD: ) which has the same symmetry group, and add a prefix indicating that the second node of the Coxeter-Dynkin Diagram should be ringed, $$t_1\{4,3\}$$. These prefixes easily allow representing any linear Coxeter-Dynkin diagram as an extended Schläfli symbol.

For convenience a number of specific prefixes have shorter names.

Regular maps
As regular maps extend the notion of regular polyhedra, some extensions for Schläfli symbols exist for encoding regular maps. Since many skew polytopes are realizations of maps in Euclidean space (or occasionally hyperbolic space) these extensions can also allow these to be encoded with extended Schläfli symbols.

Petrie polygons
Many regular maps can be written as $$\{p,q\}_r$$ indicating the polytope is $$\{p,q\}$$ with points that are $2r$ steps apart along a Petrie polygon identified. For example the Petrial cube has the symbol $$\{6,3\}_4$$ because its universal cover is $$\{6,3\}$$ and it can be obtained by identifying points which are 4 steps along its Petrie polygons, which are in this case zigzags.

When the number of steps $r$ evenly divides the size of the Petrie polygon (or it is infinite), the resulting polytope $$\{p,q\}_r$$ has a Petrie polygon with size $rr$. For clarity subscripts are never written with subscripts that don't evenly divide, so you can think of the subscript as the size of the Petrie polygon.

$t$-holes
Additional notation can be used to specify maps in terms of their "$tr$-holes". A face of a map is a path along the edges of a map always taking the rightmost (or alternatively always taking the leftmost) edge. $r$-holes generalize this notion. A $r$-hole is a path along the edges of a map such that it always takes the $r$th rightmost (or alternative always takes the $k$th leftmost) edge. Faces are thus $k$-holes. Like the extension which denotes the size of the Petrie polygon, we also can extend Schläfli symbols to denote the size of the $k$-holes. The extended Schläfli symbol $$\{p,q\mid r\}$$ indicates a symbol with Schläfli type $$\{p,q\}$$ and $k$-holes of $$\{r\}$$. Like earlier the map is constructed by taking the universal cover $$\{p,q\}$$ and identifying vertices that are distance $k$ apart along the $k$-holes. For example the mucube has the extended symbol $$\{4,6\mid 4\}$$ so it is constructed from $$\{4,6\}$$ by identifying vertices that are $1$ steps apart along the $2$-holes.

This notation can be generalized even further with multiple values after the divider. The expression $$\{p,q\mid r_2,\dots,r_n\}$$ indicates a map with type $$\{p,q\}$$ and $2$-holes of $$\{r_k\}$$ for $r$ from $2$ to $4$. Constructing these maps follows the same process as before except it is done over multiple $2$-holes.

In this notation is extended further. They prefix the size of the $k$-holes in the usual notation with a $k$ to indicate instead size of the $2$-holes in the dual map. This means that it is the size of the $n$-holes following the faces rather than the vertices. For example the hemicube is a quotient of the cube which identifies opposite faces and thus can be represented as $$\{4,3\mid *2\}$$. There is no way to represent the hemicube with the normal $$\{p,q\mid r\}$$ notation.

Hemi-polytopes
Hemi-polytopes are a quotient of spherical polytope identifying opposite points. If that spherical polytope has a Schläfli symbol, the hemi-polytope can be represented with a $k$ after its symbol. Additionally for regular hemi-polytopes the extension described above for regular maps can be used. The Schläfli symbol of the hemi-polytope is the symbol of its double cover with a subscript equal to half the number of vertices in the Petrie polygon.

Petrie dual
It may be desirable to represent the regular Petrials with Schläfli symbol. The Petrie dual of a regular polytope $$\{p,q\}$$ can be represented with a superscript pi as $$\{p,q\}^\pi$$. However this is not the only way to represent Petrials. Many Petrials can be written with the extensions for regular maps. For example the Petrial cube is $$\{6,3\}_4$$ and the Petrial tetrahedron is $$\{4,3\}_3$$. For Petrials we can generalize this to say that the symbol $$\{p,q\}_r$$ represents a polytope with $2$-gonal faces, a $$-gonal vertex figure and an $2$-gonal Petrie polygon. Thus the Petrial great stellated dodecahedron is $$\{10/3,3\}_{5/2}$$ since it's Petrie polygons are pentagrams. With this notation the Petrie dual of a polyhedron $$\{p,q\}_r$$ is $$\{r,q\}_p$$, exchanging the faces with the Petrie polygons.

Schläfli type
A Schläfli symbol $$\{p,q\}$$ indicates a regular polyhedron with $k$-gonal faces and a $/2$-gonal vertex figure. However there may be multiple abstract regular polyhedra with $p$-gonal facets and a $q$-gonal vertex figure. The idea of the Schläfli type of a polytope is to classify these polytopes with similar structures together. $$\{p,q\}$$ is the Schläfli type of all regular polyhedra with $r$-gonal faces and a $p$-gonal vertex figure although it is the Schläfli symbol for only one of them.

This idea can be made into a precise definition in a number of ways. These two definitions are equivalent.
 * 1) The Schläfli type of a polytope is the Schläfli symbol of its universal cover.
 * 2) Let  be a polytope of rank $q$, let $$\langle\rho_1,\dots,\rho_n\rangle$$ be the distinguished generators of the automorphism group of . Let $$p_i$$ be the the order of $$\rho_i\rho_{i+1}$$. The Schläfli type of  is then $$\{p_1,\dots,p_{n-1}\}$$.

In ordinary Euclidean space the Schläfli symbol and Schläfli type are always the same. That is if a polytope has a Schläfli type that is also its Schläfli symbol.