List of regular polytopes

This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.

[5] It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon.

[7] However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two.

There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}.

In general, for any natural number n, there are regular n-pointed stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m} = {n/(n − m)}) and m and n are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars).

There also exist failed star polygons, such as the piangle, which do not cover the surface of a circle finitely many times.

Existence of a regular polyhedron {p, q} is constrained by an inequality, related to the vertex figure's angle defect:

The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}: As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms.

The tiling images show a single spherical polygon face in yellow.

These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times.

Ludwig Schläfli found four of them and skipped the last six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F+V−E=2).

There are 4 unique edge arrangements and 7 unique face arrangements from these 10 regular star 4-polytopes, shown as orthogonal projections: There are 4 failed potential regular star 4-polytopes permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}.

Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.

There are two main geometric classes of apeirotope:[15] The straight apeirogon is a regular tessellation of the line, subdividing it into infinitely many equal segments.

It exists as the limit of the p-gon as p tends to infinity, as follows: Apeirogons in the hyperbolic plane, most notably the regular apeirogon, {∞}, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by horocycles or hypercycles rather than circles.

Above are two regular hyperbolic apeirogons in the Poincaré disk model, the right one shows perpendicular reflection lines of divergent fundamental domains, separated by length λ.

In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed.

It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens.

A sampling: {iπ/λ, iπ/λ} The tilings {p, ∞} have ideal vertices, on the edge of the Poincaré disc model.

Their cells and vertex figures are all regular hosohedra {2,n}, dihedra, {n,2}, and Euclidean tilings.

These improper regular tilings are constructionally related to prismatic uniform honeycombs by truncation operations.

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental tetrahedron having ultra-ideal vertices).

All honeycombs with hyperbolic cells or vertex figures and do not have 2 in their Schläfli symbol are noncompact.

Ideal vertices now appear when the vertex figure is a Euclidean tiling, becoming inscribable in a horosphere rather than a sphere.

In general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point.

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental 5-cell having some parts inaccessible beyond infinity).

There are 5 regular honeycombs in H5, all paracompact, which include infinite (Euclidean) facets or vertex figures: {3,4,3,3,3}, {3,3,4,3,3}, {3,3,3,4,3}, {3,4,3,3,4}, and {4,3,3,4,3}.

There are no regular compact or paracompact tessellations of hyperbolic space of dimension 6 or higher.

These abstract elements can be mapped into ordinary space or realised as geometrical figures.

Their construction, by arranging n faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane.