# 6-polytope

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6-dimensional geometric object

Graphs of three regular and five uniform 6-polytopes 6-simplex 6-orthoplex, 311 6-cube (Hexeract) 221 Expanded 6-simplex Rectified 6-orthoplex 6-demicube 131 (Demihexeract) 122

In [six-dimensional](/source/Six-dimensional_space) [geometry](/source/Geometry), a **six-dimensional polytope** or **6-polytope** is a [polytope](/source/Polytope), bounded by [5-polytope](/source/5-polytope) [facets](/source/Facet_(mathematics)).

## Definition

A 6-polytope is a closed six-dimensional figure with [vertices](/source/Vertex_(geometry)), [edges](/source/Edge_(geometry)), [faces](/source/Face_(geometry)), [cells](/source/Cell_(mathematics)) (3-faces), 4-faces, and 5-faces. A vertex is a [point](/source/Point_(geometry)) where six or more edges meet. An edge is a [line segment](/source/Line_segment) where four or more faces meet, and a face is a [polygon](/source/Polygon) where three or more cells meet. A cell is a [polyhedron](/source/Polyhedron). A 4-face is a [polychoron](/source/Polychoron), and a 5-face is a [5-polytope](/source/5-polytope). Furthermore, the following requirements must be met:

- Each 4-face must join exactly two 5-faces (facets).

- Adjacent facets are not in the same five-dimensional [hyperplane](/source/Hyperplane).

- The figure is not a compound of other figures which meet the requirements.

## Characteristics

The topology of any given 6-polytope is defined by its [Betti numbers](/source/Betti_number) and [torsion coefficients](/source/Torsion_coefficient_(topology)).[1]

The value of the [Euler characteristic](/source/Euler_characteristic) used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 6-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.[1]

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.[1]

## Classification

6-polytopes may be classified by properties like "[convexity](/source/Convex_set)" and "[symmetry](/source/Symmetry)".

- A 6-polytope is *[convex](/source/Convex_polytope)* if its boundary (including its 5-faces, 4-faces, cells, faces and edges) does not intersect itself and the line segment joining any two points of the 6-polytope is contained in the 6-polytope or its interior; otherwise, it is *non-convex*. Self-intersecting 6-polytope are also known as [*star 6-polytopes*](/source/Star_polytope), from analogy with the star-like shapes of the non-convex [Kepler-Poinsot polyhedra](/source/Kepler-Poinsot_polyhedra).

- A **regular 6-polytope** has all identical regular 5-polytope facets. All regular 6-polytope are convex.

Main article: [List of regular polytopes § Dimension 5 and higher](/source/List_of_regular_polytopes#Dimension_5_and_higher)

- A **[semi-regular](/source/Semiregular_polytope) 6-polytope** contains two or more types of [regular 4-polytope](/source/Regular_4-polytope) facets. There is only one such figure, called [221](/source/2_21_polytope).

- A **uniform 6-polytope** has a [symmetry group](/source/Symmetry_group) under which all vertices are equivalent, and its facets are [uniform 5-polytopes](/source/Uniform_5-polytope). The faces of a uniform polytope must be [regular](/source/Regular_polygon).

Main article: [Uniform 6-polytope](/source/Uniform_6-polytope)

- A **prismatic 6-polytope** is constructed by the [Cartesian product](/source/Cartesian_product) of two lower-dimensional polytopes. A prismatic 6-polytope is uniform if its factors are uniform. The [6-cube](/source/6-cube) is prismatic (product of a [squares](/source/Square_(geometry)) and a [cube](/source/Cube)), but is considered separately because it has symmetries other than those inherited from its factors.

- A *5-space [tessellation](/source/Tessellation)* is the division of five-dimensional [Euclidean space](/source/Euclidean_space) into a regular grid of 5-polytope facets. Strictly speaking, tessellations are not 6-polytopes as they do not bound a "6D" volume, but we include them here for the sake of completeness because they are similar in many ways to 6-polytope. A *uniform 5-space tessellation* is one whose vertices are related by a [space group](/source/Space_group) and whose facets are [uniform 5-polytopes](/source/Uniform_5-polytope).

## Regular 6-polytopes

Regular 6-polytopes can be generated from [Coxeter groups](/source/Coxeter_group) represented by the [Schläfli symbol](/source/Schl%C3%A4fli_symbol) {p,q,r,s,t} with **t** {p,q,r,s} 5-polytope [facets](/source/Facet_(mathematics)) around each [cell](/source/Cell_(geometry)).

There are only three such [convex regular 6-polytopes](/source/List_of_regular_polytopes#Convex_5):

- {3,3,3,3,3} - [6-simplex](/source/6-simplex)

- {4,3,3,3,3} - [6-cube](/source/6-cube)

- {3,3,3,3,4} - [6-orthoplex](/source/6-orthoplex)

There are no nonconvex regular polytopes of 5 or more dimensions.

For the three convex regular 6-polytopes, their elements are:

Name Schläfli symbol Coxeter diagram Vertices Edges Faces Cells 4-faces 5-faces Symmetry (order) 6-simplex {3,3,3,3,3} 7 21 35 35 21 7 A6 (720) 6-orthoplex {3,3,3,3,4} 12 60 160 240 192 64 B6 (46080) 6-cube {4,3,3,3,3} 64 192 240 160 60 12 B6 (46080)

## Uniform 6-polytopes

Main article: [Uniform 6-polytope](/source/Uniform_6-polytope)

Here are six simple uniform convex 6-polytopes, including the *6-orthoplex* repeated with its alternate construction.

Name Schläfli symbol(s) Coxeter diagram(s) Vertices Edges Faces Cells 4-faces 5-faces Symmetry (order) Expanded 6-simplex t0,5{3,3,3,3,3} 42 210 490 630 434 126 2×A6 (1440) 6-orthoplex, 311 (alternate construction) {3,3,3,31,1} 12 60 160 240 192 64 D6 (23040) 6-demicube {3,33,1} h{4,3,3,3,3} 32 240 640 640 252 44 D6 (23040) ½B6 Rectified 6-orthoplex t1{3,3,3,3,4} t1{3,3,3,31,1} 60 480 1120 1200 576 76 B6 (46080) 2×D6 221 polytope {3,3,32,1} 27 216 720 1080 648 99 E6 (51840) 122 polytope {3,32,2} or 72 720 2160 2160 702 54 2×E6 (103680)

The *expanded 6-simplex* is the [vertex figure](/source/Vertex_figure) of the uniform [6-simplex honeycomb](/source/6-simplex_honeycomb), . The [6-demicube honeycomb](/source/6-demicube_honeycomb), , vertex figure is a *rectified 6-orthoplex* and [facets](/source/Facet_(geometry)) are the *6-orthoplex* and *6-demicube*. The uniform [222 honeycomb](/source/2_22_honeycomb),, has *122* polytope is the vertex figure and *221* facets.

## References

1. ^ [***a***](#cite_ref-richeson_1-0) [***b***](#cite_ref-richeson_1-1) [***c***](#cite_ref-richeson_1-2) Richeson, D.; *Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy*, Princeton, 2008.

- [T. Gosset](/source/Thorold_Gosset): *On the Regular and Semi-Regular Figures in Space of n Dimensions*, [Messenger of Mathematics](/source/Messenger_of_Mathematics), Macmillan, 1900

- [A. Boole Stott](/source/Alicia_Boole_Stott) (1910). ["Geometrical deduction of semiregular from regular polytopes and space fillings"](https://web.archive.org/web/20250429000816/https://dwc.knaw.nl/DL/publications/PU00011492.pdf) (PDF). *Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam*. **XI** (1). Amsterdam: Johannes Müller. Archived from [the original](https://dwc.knaw.nl/DL/publications/PU00011492.pdf) (PDF) on 29 April 2025.

- [H.S.M. Coxeter](/source/Harold_Scott_MacDonald_Coxeter): - H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: *Uniform Polyhedra*, Philosophical Transactions of the Royal Society of London, 1954 - H.S.M. Coxeter, *Regular Polytopes*, 3rd edition, Dover, New York, 1973

- **Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, [Peter McMullen](/source/Peter_McMullen), Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, [wiley.com](https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036), [ISBN](/source/ISBN_(identifier)) [978-0-471-01003-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-01003-6) - (Paper 22) H.S.M. Coxeter, *Regular and Semi-Regular Polytopes I*, [Math. Zeit. 46 (1940) 380–407, MR 2,10] - (Paper 23) H.S.M. Coxeter, *Regular and Semi-Regular Polytopes II*, [Math. Zeit. 188 (1985) 559–591] - (Paper 24) H.S.M. Coxeter, *Regular and Semi-Regular Polytopes III*, [Math. Zeit. 200 (1988) 3–45]

- [N.W. Johnson](/source/Norman_Johnson_(mathematician)): *The Theory of Uniform Polytopes and Honeycombs*, Ph.D. Dissertation, University of Toronto, 1966

- Klitzing, Richard. ["6D uniform polytopes (polypeta)"](https://bendwavy.org/klitzing/dimensions/polypeta.htm).

## External links

- [Polytope names](http://www.steelpillow.com/polyhedra/ditela.html)

- [Polytopes of Various Dimensions](http://www.polytope.net/hedrondude/topes.htm), Jonathan Bowers

- [Multi-dimensional Glossary](http://tetraspace.alkaline.org/glossary.htm)

- [Glossary for hyperspace](https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html), George Olshevsky.

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

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Adapted from the Wikipedia article [6-polytope](https://en.wikipedia.org/wiki/6-polytope) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/6-polytope?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.