In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.

- Regular 10-polytopes
- Euler characteristic
- Uniform 10-polytopes by fundamental Coxeter groups
- The A10 family
- The B10 family
- The D10 family
- Regular and uniform honeycombs
- Regular and uniform hyperbolic honeycombs
- References
- External links

A **uniform 10-polytope** is one which is vertex-transitive, and constructed from uniform facets.

Regular 10-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v,w,x}, with **x** {p,q,r,s,t,u,v,w} 9-polytope facets around each peak.

There are exactly three such convex regular 10-polytopes:

- {3,3,3,3,3,3,3,3,3} - 10-simplex
- {4,3,3,3,3,3,3,3,3} - 10-cube
- {3,3,3,3,3,3,3,3,4} - 10-orthoplex

There are no nonconvex regular 10-polytopes.

The topology of any given 10-polytope is defined by its Betti numbers and torsion coefficients.^{ [1] }

The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 10-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] }

Uniform 10-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|

1 | A_{10} | [3^{9}] | |

2 | B_{10} | [4,3^{8}] | |

3 | D_{10} | [3^{7,1,1}] |

Selected regular and uniform 10-polytopes from each family include:

- Simplex family: A
_{10}[3^{9}] -- 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular:
- {3
^{9}} -**10-simplex**-

- {3

- 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular:
- Hypercube/orthoplex family: B
_{10}[4,3^{8}] -- 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones:
- {4,3
^{8}} -**10-cube**or**dekeract**- - {3
^{8},4} -**10-orthoplex**or**decacross**- - h{4,3
^{8}} -**10-demicube**.

- {4,3

- 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones:
- Demihypercube D
_{10}family: [3^{7,1,1}] -- 767 uniform 10-polytopes as permutations of rings in the group diagram, including:
**1**-_{7,1}**10-demicube**or**demidekeract**-**7**-_{1,1}**10-orthoplex**-

- 767 uniform 10-polytopes as permutations of rings in the group diagram, including:

The A_{10} family has symmetry of order 39,916,800 (11 factorial).

There are 512+16-1=527 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. 31 are shown below: all one and two ringed forms, and the final omnitruncated form. Bowers-style acronym names are given in parentheses for cross-referencing.

# | Graph | Coxeter-Dynkin diagram Schläfli symbol Name | Element counts | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

9-faces | 8-faces | 7-faces | 6-faces | 5-faces | 4-faces | Cells | Faces | Edges | Vertices | |||

1 |
| 11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 | |

2 |
| 495 | 55 | |||||||||

3 |
| 1980 | 165 | |||||||||

4 |
| 4620 | 330 | |||||||||

5 |
| 6930 | 462 | |||||||||

6 |
| 550 | 110 | |||||||||

7 |
| 4455 | 495 | |||||||||

8 |
| 2475 | 495 | |||||||||

9 |
| 15840 | 1320 | |||||||||

10 |
| 17820 | 1980 | |||||||||

11 |
| 6600 | 1320 | |||||||||

12 |
| 32340 | 2310 | |||||||||

13 |
| 55440 | 4620 | |||||||||

14 |
| 41580 | 4620 | |||||||||

15 |
| 11550 | 2310 | |||||||||

16 |
| 41580 | 2772 | |||||||||

17 |
| 97020 | 6930 | |||||||||

18 |
| 110880 | 9240 | |||||||||

19 |
| 62370 | 6930 | |||||||||

20 |
| 13860 | 2772 | |||||||||

21 |
| 34650 | 2310 | |||||||||

22 |
| 103950 | 6930 | |||||||||

23 |
| 161700 | 11550 | |||||||||

24 |
| 138600 | 11550 | |||||||||

25 |
| 18480 | 1320 | |||||||||

26 |
| 69300 | 4620 | |||||||||

27 |
| 138600 | 9240 | |||||||||

28 |
| 5940 | 495 | |||||||||

29 |
| 27720 | 1980 | |||||||||

30 |
| 990 | 110 | |||||||||

31 | t _{0,1,2,3,4,5,6,7,8,9}{3,3,3,3,3,3,3,3,3}Omnitruncated 10-simplex | 199584000 | 39916800 |

There are 1023 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

Twelve cases are shown below: ten single-ring (rectified) forms, and two truncations. Bowers-style acronym names are given in parentheses for cross-referencing.

# | Graph | Coxeter-Dynkin diagram Schläfli symbol Name | Element counts | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

9-faces | 8-faces | 7-faces | 6-faces | 5-faces | 4-faces | Cells | Faces | Edges | Vertices | |||

1 | t _{0}{4,3,3,3,3,3,3,3,3}10-cube (deker) | 20 | 180 | 960 | 3360 | 8064 | 13440 | 15360 | 11520 | 5120 | 1024 | |

2 | t _{0,1}{4,3,3,3,3,3,3,3,3}Truncated 10-cube (tade) | 51200 | 10240 | |||||||||

3 | t _{1}{4,3,3,3,3,3,3,3,3}Rectified 10-cube (rade) | 46080 | 5120 | |||||||||

4 | t _{2}{4,3,3,3,3,3,3,3,3}Birectified 10-cube (brade) | 184320 | 11520 | |||||||||

5 | t _{3}{4,3,3,3,3,3,3,3,3}Trirectified 10-cube (trade) | 322560 | 15360 | |||||||||

6 | t _{4}{4,3,3,3,3,3,3,3,3}Quadrirectified 10-cube (terade) | 322560 | 13440 | |||||||||

7 | t _{4}{3,3,3,3,3,3,3,3,4}Quadrirectified 10-orthoplex (terake) | 201600 | 8064 | |||||||||

8 | t _{3}{3,3,3,3,3,3,3,4}Trirectified 10-orthoplex (trake) | 80640 | 3360 | |||||||||

9 | t _{2}{3,3,3,3,3,3,3,3,4}Birectified 10-orthoplex (brake) | 20160 | 960 | |||||||||

10 | t _{1}{3,3,3,3,3,3,3,3,4}Rectified 10-orthoplex (rake) | 2880 | 180 | |||||||||

11 | t _{0,1}{3,3,3,3,3,3,3,3,4}Truncated 10-orthoplex (take) | 3060 | 360 | |||||||||

12 | t _{0}{3,3,3,3,3,3,3,3,4}10-orthoplex (ka) | 1024 | 5120 | 11520 | 15360 | 13440 | 8064 | 3360 | 960 | 180 | 20 |

The D_{10} family has symmetry of order 1,857,945,600 (10 factorial × 2^{9}).

This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D_{10} Coxeter-Dynkin diagram. Of these, 511 (2×256−1) are repeated from the B_{10} family and 256 are unique to this family, with 2 listed below. Bowers-style acronym names are given in parentheses for cross-referencing.

# | Graph | Coxeter-Dynkin diagram Schläfli symbol Name | Element counts | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

9-faces | 8-faces | 7-faces | 6-faces | 5-faces | 4-faces | Cells | Faces | Edges | Vertices | |||

1 | 10-demicube (hede) | 532 | 5300 | 24000 | 64800 | 115584 | 142464 | 122880 | 61440 | 11520 | 512 | |

2 | Truncated 10-demicube (thede) | 195840 | 23040 |

There are four fundamental affine Coxeter groups that generate regular and uniform tessellations in 9-space:

# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|

1 | [3^{[10]}] | ||

2 | [4,3^{7},4] | ||

3 | h[4,3^{7},4][4,3 ^{6},3^{1,1}] | ||

4 | q[4,3^{7},4][3 ^{1,1},3^{5},3^{1,1}] |

Regular and uniform tessellations include:

- Regular 9-hypercubic honeycomb, with symbols {4,3
^{7},4}, - Uniform alternated 9-hypercubic honeycomb with symbols h{4,3
^{7},4},

There are no compact hyperbolic Coxeter groups of rank 10, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.

= [3^{1,1},3^{4},3^{2,1}]: | = [4,3^{5},3^{2,1}]: | or = [3^{6,2,1}]: |

Three honeycombs from the family, generated by end-ringed Coxeter diagrams are:

In five-dimensional geometry, a **five-dimensional polytope** or **5-polytope** is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

In eight-dimensional geometry, an **eight-dimensional polytope** or **8-polytope** is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

In seven-dimensional geometry, a **7-polytope** is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

In nine-dimensional geometry, a **nine-dimensional polytope** or **9-polytope** is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.

In six-dimensional geometry, a **uniform polypeton** is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

The **5-demicube honeycomb** is a uniform space-filling tessellation in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.

In geometry, a **uniform 5-polytope** is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

In 6-dimensional geometry, the **1 _{22}** polytope is a uniform polytope, constructed from the E

In 7-dimensional geometry, **1 _{32}** is a uniform polytope, constructed from the E7 group.

In 8-dimensional geometry, the **1 _{42}** is a uniform 8-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **2 _{41}** is a uniform 8-polytope, constructed within the symmetry of the E

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

In 7-dimensional geometry, the **3 _{21}** polytope is a uniform 7-polytope, constructed within the symmetry of the E

In six-dimensional geometry, a **pentellated 6-simplex** is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

In six-dimensional geometry, a **six-dimensional polytope** or **6-polytope** is a polytope, bounded by 5-polytope facets.

In six-dimensional geometry, a **rectified 6-simplex** is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

In six-dimensional geometry, a **truncated 6-simplex** is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

In eight-dimensional geometry, a **rectified 8-simplex** is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

In six-dimensional geometry, a **runcinated 6-simplex** is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex.

In geometry, an **E _{9} honeycomb** is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E

- T. Gosset:
*On the Regular and Semi-Regular Figures in Space of n Dimensions*, Messenger of Mathematics, Macmillan, 1900 - A. Boole Stott:
*Geometrical deduction of semiregular from regular polytopes and space fillings*, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910 - H.S.M. Coxeter:
- H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller:
*Uniform Polyhedra*, Philosophical Transactions of the Royal Society of London, Londne, 1954 - H.S.M. Coxeter,
*Regular Polytopes*, 3rd Edition, Dover New York, 1973

- H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller:
**Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 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]

- (Paper 22) H.S.M. Coxeter,
- N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D. Dissertation, University of Toronto, 1966 - Klitzing, Richard. "10D uniform polytopes (polyxenna)".

- Polytope names
- Polytopes of Various Dimensions, Jonathan Bowers
- Multi-dimensional Glossary
- Glossary for hyperspace , George Olshevsky.

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