Schläfli symbol
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli,[1]: 143 who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions.
For example, {3} is an equilateral triangle, {4} is a square, {5} a convex regular pentagon, etc.
Regular star polygons are not convex, and their Schläfli symbols
Regular polytopes can have star polygon elements, like the pentagram, with symbol {5⁄2}, represented by the vertices of a pentagon but connected alternately.
The Schläfli symbol can represent a finite convex polyhedron, an infinite tessellation of Euclidean space, or an infinite tessellation of hyperbolic space, depending on the angle defect of the construction.
A positive angle defect allows the vertex figure to fold into a higher dimension and loops back into itself as a polytope.
A zero angle defect tessellates space of the same dimension as the facets.
Usually, a facet or a vertex figure is assumed to be a finite polytope, but can sometimes itself be considered a tessellation.
A self-dual regular polytope will have a symmetric Schläfli symbol.
[1]: 138 Schläfli's work was almost unknown in his lifetime, and his notation for describing polytopes was rediscovered independently by several others.
[1]: 144 Gosset's form has greater symmetry, so the number of dimensions is the number of vertical bars, and the symbol exactly includes the sub-symbols for facet and vertex figure.
Gosset regarded | p as an operator, which can be applied to | q | ... | z | to produce a polytope with p-gonal faces whose vertex figure is | q | ... | z |.
Schläfli symbols are closely related to (finite) reflection symmetry groups, which correspond precisely to the finite Coxeter groups and are specified with the same indices, but square brackets instead [p,q,r,...].
The Schläfli symbol of a convex regular polygon with p edges is {p}.
See the 5 convex Platonic solids, the 4 nonconvex Kepler-Poinsot polyhedra.
Topologically, a regular 2-dimensional tessellation may be regarded as similar to a (3-dimensional) polyhedron, but such that the angular defect is zero.
Thus, Schläfli symbols may also be defined for regular tessellations of Euclidean or hyperbolic space in a similar way as for polyhedra.
If a 4-polytope's symbol is palindromic (e.g. {3,3,3} or {3,4,3}), its bitruncation will only have truncated forms of the vertex figure as cells.
The prismatic duals, or bipyramids can be represented as composite symbols, but with the addition operator, "+".
Pyramidal polytopes containing vertices orthogonally offset can be represented using a join operator, "∨".
Every pair of vertices between joined figures are connected by edges.
Axial polytopes containing vertices on parallel offset hyperplanes can be represented by the ‖ operator.
An altered even-sided regular 2n-gon generates a star figure compound, 2{n}.
Norman Johnson simplified the notation for vertical symbols with an r prefix.
The t-notation is the most general, and directly corresponds to the rings of the Coxeter diagram.
Symbols have a corresponding alternation, replacing rings with holes in a Coxeter diagram and h prefix standing for half, construction limited by the requirement that neighboring branches must be even-ordered and cuts the symmetry order in half.
A related operator, a for altered, is shown with two nested holes, represents a compound polyhedra with both alternated halves, retaining the original full symmetry.
Alternations have half the symmetry of the Coxeter groups and are represented by unfilled rings.
Quarter forms are shown here with a + inside a hollow ring to imply they are two independent alternations.
