User:Mike40033/List of regular polytopes

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

The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.

The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one lower dimensional Euclidean space.

Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with 7 equilateral triangles and allowing it to lie flat. It can't be done in a regular plane, but can be at the right scale of a hyperbolic plane.

Two-dimensional regular polytopes
The two dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic.

Usually only convex polygons are considered regular, but star polygons, like the pentagram, can also be considered regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to complete.

Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.

Three-dimensional regular polytopes
In three dimensions, the regular polytopes are called polyhedra:

A regular polyhedron with Schläfli symbol {p,q} has a regular face type {p}, and regular vertex figure {q}.

A polyhedral vertex figure is an imaginary polygon which can be seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.

Existence of a regular polyhedron {p,q} is constrained by an inequality, related to the vertex figure's angle defect:
 * 1/p + 1/q > 1/2 : Polyhedron (existing in Euclidean 3-space)
 * 1/p + 1/q = 1/2 : Euclidean plane tiling
 * 1/p + 1/q < 1/2 : Hyperbolic plane tiling

By enumerating the permutations, we find 6 convex forms, 10 nonconvex forms and 3 plane tilings, all with polygons {p} and {q} limited to: {3}, {4}, {5}, {5/2}, and {6}.

Beyond Euclidean space, there's an infinite set of regular hyperbolic tilings.

Four-dimensional regular polytopes
Regular polychora with Schläfli symbol symbol {p,q,r} have cells of type {p,q}, faces of type {p}, edge figures {r}, and vertex figures {q,r}.
 * A polychoral vertex figure is an imaginary polyhedron that can be seen by the arrangement of neighboring vertices around a given vertex. For regular polychora, this vertex figure is a regular polyhedron.
 * A polychoral edge figure is an imaginary polygon that can be seen by the arrangement of faces around an edge. For regular polychora, this edge figure will always be a regular polygon.

The existence of a regular polychoron {p,q,r} is constrained by the existence of the regular polyhedra {p,q}, {q,r}.

Each will exist in a space dependent upon this expression:
 * sin(&pi;/p) sin(&pi;/r) &minus; cos(&pi;/q)
 * > 0 : Hyperspherical surface polychoron (in 4-space)
 * = 0 : Euclidean 3-space honeycomb
 * < 0 : Hyperbolic 3-space honeycomb

These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.

The Euler characteristic &chi; for polychora is &chi; = V + F &minus; E &minus; C and is zero for all forms.

Five-dimensional regular polytopes
In five dimensions, a regular polytope can be named as {p,q,r,s} where {p,q,r} is the hypercell (or teron) type, {p,q} is the cell type, {p} is the face type, and {s} is the face figure, {r,s} is the edge figure, and {q,r,s} is the vertex figure.

A 5-polytope has been called a polyteron, and if infinite (i.e. a honeycomb) a 5-polytope can be called a tetracomb.


 * A polyteron vertex figure is an imaginary polychoron that can be seen by the arrangement of neighboring vertices to each vertex.
 * A polyteron edge figure is an imaginary polyhedron that can be seen by the arrangement of faces around each edge.
 * A polyteron face figure is an imaginary polygon that can be seen by the arrangement of cells around each face.

A regular polytope {p,q,r,s} exists only if {p,q,r} and {q,r,s} are regular polychora.

The space it fits in is based on the expression:
 * (cos2(&pi;/q)/sin2(&pi;/p)) + (cos2(&pi;/r)/sin2(&pi;/s))
 * < 1 : Spherical polytope
 * = 1 : Euclidean 4-space tessellation
 * > 1 : hyperbolic 4-space tessellation

Enumeration of these constraints produce 3 convex polytopes, zero nonconvex polytopes, 3 4-space tessellations, and 5 hyperbolic 4-space tessellations.

Two Dimensions
The Schläfli symbol {p} represents a regular p-gon:

The infinite set of convex regular polygons are:

A digon, {2}, can be considered a degenerate regular polygon.

Three Dimensions
The convex regular polyhedra are called the 5 Platonic solids:

In spherical geometry, hosohedron, {2,n} and dihedron {n,2} can be considered regular polyhedra (tilings of the sphere).

Four Dimensions
The 6 convex polychora are as follows:

Five Dimensions
There are three kinds of convex regular polytopes in five dimensions:

Higher dimensions
In dimensions 5 and higher, there are only three kinds of convex regular polytopes.

Two Dimensions
There exist non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers: star polygons.

In general, for any natural number n, there are n-pointed non-convex regular polygonal 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.

Three Dimensions
The nonconvex regular polyhedra are call the Kepler-Poinsot solids and there are four of them, based on the vertices of the dodecahedron {5,3} and icosahedron {3,5}:



Four Dimensions
There are ten nonconvex regular polychora, which can be called Schläfli-Hess polychora and their vertices are based on the convex 120-cell {5,3,3} and 600-cell {3,3,5}:

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 toruses: F+V-E=2). Edmund Hess (1843-1903) completed the full list of ten in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder.

There are 4 failed potential nonconvex regular polychora 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.

Higher dimensions
There are no non-convex regular polytopes in five dimension or higher.

Tesselations
The classical convex polytopes may be considered tesselations, or tilings of spherical space. Tesselations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tesselates a space of one dimension less. For example, the (three dimensional) platonic solids tesselate the 'two'-dimensional 'surface' of the sphere.

Two dimensions
There is one tesselation of the line, giving one polytope, the (two-dimensional) apeirogon. This has infinitely many vertices and edges. It's Schläfli is {&infin;}.

Euclidean (Plane) Tilings
There are three regular tesselations of the plane.

Euclidean star-tilings
There are no plane tilings of star polygons. There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc, but none repeat periodically.

Hyperbolic tilings
Tessellations of hyperbolic 2-space can be called hyperbolic tilings.

There are infinitely many regular hyperbolic tilings. As stated above, every positive integer pairs {p,q} such that 1/p + 1/q < 1/2 is a hyperbolic tiling.

There are 2 forms of star-tilings {m/2, m} and their duals {m,m/2} with m=3,5,7,...

A sampling:

There's a number of different ways to display the hyperbolic plane, including the Poincaré disc model below which maps the plane into a circle, as shown below. 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.

Tesselations of Euclidean 3-space
Tessellations of 3-space are called honeycombs. There is only one regular honeycomb:

Tesselations of Hyperbolic 3-space
Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs. There are 4 regular hyperbolic honeycombs:

There are also 11 other H3 honeycombs which have infinite cells and/or infinite vertex figures: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, {6,3,6}.

Tesselations of Euclidean 4-space
There are three kinds of infinite regular tessellations (tetracombs) that can tessellate four dimensional space:

Tesselations of Hyperbolic 4-space
There are five kinds of convex regular tetracombs and four kinds of star-honeycombs in H4 space.

There are also 2 more H4 honeycombs with infinite facets or vertex figures: {3,4,3,4}, {4,3,4,3}

Tesselations of Euclidean Space
There is only one infinite regular polytope that can tessellate five dimensions or higher, formed by measure polytopes (that is, 'n'-dimensional cubes).

Tesselations of Hyperbolic Space
There are no finite-faceted regular tessellations of hyperbolic space of dimension 5 or higher.

There are 5 regular honeycombs in H5 with infinite facets or vertex figures: {3,3,3,4,3}, {3,4,3,3,3}, {3,3,4,3,3}, {3,4,3,3,4}, {4,3,3,4,3}.

Two dimensions
The apeirogon has regular embeddings in the plane, and in higher-dimensional spaces. In two dimensions it forms a zig-zag. In three dimensions, it traces out a helical spiral.

Three dimensions
There are thirty regular apeirohedra in Euclidean space. See section 7E of Abstract Regular Polytopes, by McMullen and Schulte. These include the tesselations of type {4,4}, {6,3} and {3,6} above, as well as (in the plane) polytopes of type {&infin;,3}, {&infin;,4} and {&infin;,6}, and in 3-dimensional space, blends of these with either an apeirogon or a line segment, and the "pure" 3-dimensional apeirohedra (12 in number)

Four and higher dimensions
The apeirochora have not been completely classified as of 2006.

Abstract Polytopes
The abstract polytopes arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. The include the tesselations of spherical, euclidean and hyperbolic space, tesselations of other manifolds, and many other objects that do not have a well-defined topology, but instead may be characterised by their "local" topology. There are infinitely many in every dimension. See this atlas for a sample.