# Uniform 10-polytope > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Uniform_10-polytope > Markdown URL: https://mediated.wiki/source/Uniform_10-polytope.md > Source: https://en.wikipedia.org/wiki/Uniform_10-polytope > Source revision: 1356715483 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Type of geometrical object Graphs of three regular and related uniform polytopes. 10-simplex Truncated 10-simplex Rectified 10-simplex Cantellated 10-simplex Runcinated 10-simplex Stericated 10-simplex Pentellated 10-simplex Hexicated 10-simplex Heptellated 10-simplex Octellated 10-simplex Ennecated 10-simplex 10-orthoplex Truncated 10-orthoplex Rectified 10-orthoplex 10-cube Truncated 10-cube Rectified 10-cube 10-demicube Truncated 10-demicube In ten-dimensional [geometry](/source/Geometry), a 10-polytope is a 10-dimensional [polytope](/source/Polytope) whose boundary consists of [9-polytope](/source/9-polytope) [facets](/source/Facet_(mathematics)), exactly two such facets meeting at each [8-polytope](/source/8-polytope) [ridge](/source/Ridge_(geometry)). A **uniform 10-polytope** is one which is [vertex-transitive](/source/Vertex-transitive), and constructed from [uniform](/source/Uniform_9-polytope) [facets](/source/Facet_(geometry)). ## Regular 10-polytopes Regular 10-polytopes can be represented by the [Schläfli symbol](/source/Schl%C3%A4fli_symbol) {p,q,r,s,t,u,v,w,x}, with **x** {p,q,r,s,t,u,v,w} 9-polytope [facets](/source/Facet_(mathematics)) around each [peak](/source/Peak_(geometry)). There are exactly three such [convex regular 10-polytopes](/source/List_of_regular_polytopes#Convex_4): 1. {3,3,3,3,3,3,3,3,3} - [10-simplex](/source/10-simplex) 1. {4,3,3,3,3,3,3,3,3} - [10-cube](/source/10-cube) 1. {3,3,3,3,3,3,3,3,4} - [10-orthoplex](/source/10-orthoplex) There are no nonconvex regular 10-polytopes. ## Euler characteristic The topology of any given 10-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 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 by fundamental Coxeter groups Uniform 10-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the [Coxeter-Dynkin diagrams](/source/Coxeter-Dynkin_diagram): # Coxeter group Coxeter-Dynkin diagram 1 A10 [39] 2 B10 [4,38] 3 D10 [37,1,1] Selected regular and uniform 10-polytopes from each family include: 1. [Simplex](/source/Simplex) family: A10 [39] - - 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular: 1. {39} - **[10-simplex](/source/10-simplex)** - 1. [Hypercube](/source/Hypercube)/[orthoplex](/source/Orthoplex) family: B10 [4,38] - - 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones: 1. {4,38} - **[10-cube](/source/10-cube)** or **dekeract** - 1. {38,4} - **[10-orthoplex](/source/10-orthoplex)** or **decacross** - 1. h{4,38} - **[10-demicube](/source/10-demicube)** - 1. [Demihypercube](/source/Demihypercube) D10 family: [37,1,1] - - 767 uniform 10-polytopes as permutations of rings in the group diagram, including: 1. **17,1** - **[10-demicube](/source/10-demicube)** or **demidekeract** - 1. **71,1** - **[10-orthoplex](/source/10-orthoplex)** - ## The A10 family The A10 family has symmetry of order 39,916,800 (11 [factorial](/source/Factorial)). There are 512+16-1=527 forms based on all permutations of the [Coxeter-Dynkin diagrams](/source/Coxeter-Dynkin_diagram) 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 t0{3,3,3,3,3,3,3,3,3} 10-simplex (ux) 11 55 165 330 462 462 330 165 55 11 2 t1{3,3,3,3,3,3,3,3,3} Rectified 10-simplex (ru) 495 55 3 t2{3,3,3,3,3,3,3,3,3} Birectified 10-simplex (bru) 1980 165 4 t3{3,3,3,3,3,3,3,3,3} Trirectified 10-simplex (tru) 4620 330 5 t4{3,3,3,3,3,3,3,3,3} Quadrirectified 10-simplex (teru) 6930 462 6 t0,1{3,3,3,3,3,3,3,3,3} Truncated 10-simplex (tu) 550 110 7 t0,2{3,3,3,3,3,3,3,3,3} Cantellated 10-simplex 4455 495 8 t1,2{3,3,3,3,3,3,3,3,3} Bitruncated 10-simplex 2475 495 9 t0,3{3,3,3,3,3,3,3,3,3} Runcinated 10-simplex 15840 1320 10 t1,3{3,3,3,3,3,3,3,3,3} Bicantellated 10-simplex 17820 1980 11 t2,3{3,3,3,3,3,3,3,3,3} Tritruncated 10-simplex 6600 1320 12 t0,4{3,3,3,3,3,3,3,3,3} Stericated 10-simplex 32340 2310 13 t1,4{3,3,3,3,3,3,3,3,3} Biruncinated 10-simplex 55440 4620 14 t2,4{3,3,3,3,3,3,3,3,3} Tricantellated 10-simplex 41580 4620 15 t3,4{3,3,3,3,3,3,3,3,3} Quadritruncated 10-simplex 11550 2310 16 t0,5{3,3,3,3,3,3,3,3,3} Pentellated 10-simplex 41580 2772 17 t1,5{3,3,3,3,3,3,3,3,3} Bistericated 10-simplex 97020 6930 18 t2,5{3,3,3,3,3,3,3,3,3} Triruncinated 10-simplex 110880 9240 19 t3,5{3,3,3,3,3,3,3,3,3} Quadricantellated 10-simplex 62370 6930 20 t4,5{3,3,3,3,3,3,3,3,3} Quintitruncated 10-simplex 13860 2772 21 t0,6{3,3,3,3,3,3,3,3,3} Hexicated 10-simplex 34650 2310 22 t1,6{3,3,3,3,3,3,3,3,3} Bipentellated 10-simplex 103950 6930 23 t2,6{3,3,3,3,3,3,3,3,3} Tristericated 10-simplex 161700 11550 24 t3,6{3,3,3,3,3,3,3,3,3} Quadriruncinated 10-simplex 138600 11550 25 t0,7{3,3,3,3,3,3,3,3,3} Heptellated 10-simplex 18480 1320 26 t1,7{3,3,3,3,3,3,3,3,3} Bihexicated 10-simplex 69300 4620 27 t2,7{3,3,3,3,3,3,3,3,3} Tripentellated 10-simplex 138600 9240 28 t0,8{3,3,3,3,3,3,3,3,3} Octellated 10-simplex 5940 495 29 t1,8{3,3,3,3,3,3,3,3,3} Biheptellated 10-simplex 27720 1980 30 t0,9{3,3,3,3,3,3,3,3,3} Ennecated 10-simplex 990 110 31 t0,1,2,3,4,5,6,7,8,9{3,3,3,3,3,3,3,3,3} Omnitruncated 10-simplex 199584000 39916800 ## The B10 family There are 1023 forms based on all permutations of the [Coxeter-Dynkin diagrams](/source/Coxeter-Dynkin_diagram) with one or more rings. Twelve cases are shown below: ten single-ring ([rectified](/source/Rectification_(geometry))) 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 t0{4,3,3,3,3,3,3,3,3} 10-cube (deker) 20 180 960 3360 8064 13440 15360 11520 5120 1024 2 t0,1{4,3,3,3,3,3,3,3,3} Truncated 10-cube (tade) 51200 10240 3 t1{4,3,3,3,3,3,3,3,3} Rectified 10-cube (rade) 46080 5120 4 t2{4,3,3,3,3,3,3,3,3} Birectified 10-cube (brade) 184320 11520 5 t3{4,3,3,3,3,3,3,3,3} Trirectified 10-cube (trade) 322560 15360 6 t4{4,3,3,3,3,3,3,3,3} Quadrirectified 10-cube (terade) 322560 13440 7 t4{3,3,3,3,3,3,3,3,4} Quadrirectified 10-orthoplex (terake) 201600 8064 8 t3{3,3,3,3,3,3,3,4} Trirectified 10-orthoplex (trake) 80640 3360 9 t2{3,3,3,3,3,3,3,3,4} Birectified 10-orthoplex (brake) 20160 960 10 t1{3,3,3,3,3,3,3,3,4} Rectified 10-orthoplex (rake) 2880 180 11 t0,1{3,3,3,3,3,3,3,3,4} Truncated 10-orthoplex (take) 3060 360 12 t0{3,3,3,3,3,3,3,3,4} 10-orthoplex (ka) 1024 5120 11520 15360 13440 8064 3360 960 180 20 ## The D10 family The D10 family has symmetry of order 1,857,945,600 (10 [factorial](/source/Factorial) × 29). This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D10 [Coxeter-Dynkin diagram](/source/Coxeter-Dynkin_diagram). Of these, 511 (2×256−1) are repeated from the B10 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 ## Regular and uniform honeycombs There are four fundamental affine [Coxeter groups](/source/Coxeter_groups) that generate regular and uniform tessellations in 9-space: # Coxeter group Coxeter-Dynkin diagram 1 A ~ 9 {\displaystyle {\tilde {A}}_{9}} [3[10]] 2 B ~ 9 {\displaystyle {\tilde {B}}_{9}} [4,37,4] 3 C ~ 9 {\displaystyle {\tilde {C}}_{9}} h[4,37,4] [4,36,31,1] 4 D ~ 9 {\displaystyle {\tilde {D}}_{9}} q[4,37,4] [31,1,35,31,1] Regular and uniform tessellations include: - [Regular](/source/List_of_regular_polytopes#Higher_dimensions_3) [9-hypercubic honeycomb](/source/Hypercubic_honeycomb), with symbols {4,37,4}, - Uniform [alternated 9-hypercubic honeycomb](/source/Alternated_hypercubic_honeycomb) with symbols h{4,37,4}, ### Regular and uniform hyperbolic honeycombs There are no compact hyperbolic Coxeter groups of rank 10, groups that can generate honeycombs with all finite facets, and a finite [vertex figure](/source/Vertex_figure). However, there are [3 paracompact hyperbolic Coxeter groups](/source/Coxeter-Dynkin_diagram#Rank_4_to_10) of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams. Q ¯ 9 {\displaystyle {\bar {Q}}_{9}} = [31,1,34,32,1]: S ¯ 9 {\displaystyle {\bar {S}}_{9}} = [4,35,32,1]: E 10 {\displaystyle E_{10}} or T ¯ 9 {\displaystyle {\bar {T}}_{9}} = [36,2,1]: Three honeycombs from the E 10 {\displaystyle E_{10}} family, generated by end-ringed Coxeter diagrams are: - [621 honeycomb](/source/6_21_honeycomb): - [261 honeycomb](/source/2_61_honeycomb): - [162 honeycomb](/source/1_62_honeycomb): ## 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, Londne, 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, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, [ISBN](/source/ISBN_(identifier)) [978-0-471-01003-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-01003-6) [\[1\]](http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html) [Archived](https://web.archive.org/web/20160711140441/http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html) 2016-07-11 at the [Wayback Machine](/source/Wayback_Machine) - (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. ["10D uniform polytopes (polyxenna)"](https://bendwavy.org/klitzing/dimensions/polyxenna.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 --- Adapted from the Wikipedia article [Uniform 10-polytope](https://en.wikipedia.org/wiki/Uniform_10-polytope) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Uniform_10-polytope?action=history)). 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