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 cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.

Regular polytope summary count by dimension

A henagon {1}, and digon {2}, can be considered a degenerate regular polygon.

Three dimensions

The convex regular polyhedra are called the 5 Platonic solids. (The vertex figure is given with each vertex count.)

Finite non-convex polytopes - star-polytopes

Two dimensions

There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {m/n}. They are called 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.

Four dimensions

There are ten regular star 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" [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001] .

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.

Four dimensions

Tessellations of Euclidean 3-space

There is only one regular tessellation of 3-space ("honeycombs"):

There are also 11 H3 honeycombs which have infinite (Euclidean) cells and/or 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}.

Five dimensions

Tessellations of Euclidean 4-space

There are three kinds of infinite regular tessellations (honeycombs) that can tessellate four dimensional space:

Four regular star-honeycombs in H⁴ space:

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

Higher dimensions

Tessellations of Euclidean Space

The hypercube honeycomb is the only family of regular honeycombs that can tessellate each dimension, five or higher, formed by hypercube facets, four around every ridge.

Tessellations 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 (Euclidean) facets or vertex figures:$\{3,4,3,3,3\},\; \{3,3,4,3,3\},\; \{3,3,3,4,3\},\; \{3,4,3,3,4\},\; \{4,3,3,4,3\}$.

Even allowing for infinite (Euclidean) facets and/or vertex figures, there are no regular tessellations of hyperbolic space of dimension 6 or higher.

Apeirotopes

An apeirotope is, like any other polytope, an unbounded hyper-surface. The difference is that whereas a polytope's hyper-surface curls back on itself to close round a finite volume of hyperspace, an apeirotope just goes on for ever.

Some people regard apeirotopes as just a special kind of polytope, while others regard them as rather different things.

Two dimensions

A regular apeirogon is a regular division of an infinitely long line into equal segments, joined by vertices. It has regular embeddings in the plane, and in higher-dimensional spaces. In two dimensions it can form a straight line or a zig-zag. In three dimensions, it traces out a helical spiral. The zig-zag and spiral forms are said to be skew.

Three dimensions

An apeirohedron is an infinite polyhedral surface. Like an apeirogon, it can be flat or skew. A flat apeirohedron is just a tiling of the plane. A skew apeirohedron is an intricate honeycomb-like structure which divides space into two regions.

There are thirty regular apeirohedra in Euclidean space. See section 7E of Abstract Regular Polytopes, by McMullen and Schulte. These include the tessellations of type$\{4,4\},\; \{6,3\}$ and $\{3,6\}$ above, as well as (in the plane) polytopes of type:$\{infty,3\}$, $\{infty,4\}$ and $\{infty,6\}$, 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. They include the tessellations of spherical, euclidean and hyperbolic space, tessellations 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 [http://www.abstract-polytopes.com/atlas/ this atlas] for a sample. Some notable examples of abstract polytopes that do not appear elsewhere in this list are the 11-cell and the 57-cell.

See also

* Polygon
** Regular polygon
** Star polygon
* Polyhedron
** Regular polyhedron (5 regular Platonic solids and 4 Kepler-Poinsot solids)
*** Uniform polyhedron
* Polychoron
** Convex regular 4-polytope (6 regular polychora)
*** Uniform polychoron
** Schläfli-Hess polychoron (10 regular star polychora)
* Tessellation
** Tilings of regular polygons
** Convex uniform honeycomb
* Regular polytope
** Uniform polytope

References

*Coxeter, "Regular Polytopes", 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294-296)
*Coxeter, "The Beauty of Geometry: Twelve Essays", Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
* D. M. Y. Sommerville, "An Introduction to the Geometry of n Dimensions." New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes

External links

* [http://www.math.utah.edu/~alfeld/math/polyhedra/polyhedra.html The Platonic Solids]
* [http://www.georgehart.com/virtual-polyhedra/kepler-poinsot-info.html Kepler-Poinsot Polyhedra]
* [http://www.weimholt.com/andrew/polytope.shtml Regular 4d Polytope Foldouts]
* [http://members.aol.com/Polycell/glossary.html#H Multidimensional Glossary] (Look up Hexacosichoron and Hecatonicosachoron)
* [http://www.geocities.com/shapirojon34/tesseract/TesseractApplet.html Polytope Viewer]
* [http://presh.com/hovinga/ Polytopes and optimal packing of p points in n dimensional spheres]
* [http://www.abstract-polytopes.com/atlas/ An atlas of small regular polytopes]