+ Graphs of three regular and related uniform polytopes.10-simplexTruncated 10-simplexRectified 10-simplexCantellated 10-simplexRuncinated 10-simplexStericated 10-simplexPentellated 10-simplexHexicated 10-simplexHeptellated 10-simplexOctellated 10-simplexEnnecated 10-simplex10-orthoplexTruncated 10-orthoplexRectified 10-orthoplex10-cubeTruncated 10-cubeRectified 10-cube10-demicubeTruncated 10-demicube 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.
A uniform 10-polytope is one which is vertex-transitive, and constructed from uniform facets.
Regular 10-polytopes 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.
Euler characteristic The topology of any given 10-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
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.
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.
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:
#Coxeter groupCoxeter-Dynkin diagram1A10 2B10[4,38]3D10[37,1,1]
Selected regular and uniform 10-polytopes from each family include: Simplex family: A10 - 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular: {39} - 10-simplex - Hypercube/orthoplex family: B10 [4,38] - 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones: {4,38} - 10-cube or dekeract - {38,4} - 10-orthoplex or decacross - h{4,38} - 10-demicube - Demihypercube D10 family: [37,1,1] - 767 uniform 10-polytopes as permutations of rings in the group diagram, including: 17,1 - 10-demicube or demidekeract - 71,1 - 10-orthoplex -
The A10 family The A10 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.
#GraphCoxeter-Dynkin diagramSchläfli symbolNameElement counts 9-faces 8-faces 7-faces 6-faces 5-faces 4-faces Cells Faces Edges Vertices1t0{3,3,3,3,3,3,3,3,3}10-simplex (ux)115516533046246233016555112t1{3,3,3,3,3,3,3,3,3}Rectified 10-simplex (ru) 495 553t2{3,3,3,3,3,3,3,3,3}Birectified 10-simplex (bru) 1980 1654t3{3,3,3,3,3,3,3,3,3}Trirectified 10-simplex (tru) 4620 3305t4{3,3,3,3,3,3,3,3,3}Quadrirectified 10-simplex (teru) 6930 4626t0,1{3,3,3,3,3,3,3,3,3}Truncated 10-simplex (tu) 550 1107t0,2{3,3,3,3,3,3,3,3,3}Cantellated 10-simplex 4455 4958t1,2{3,3,3,3,3,3,3,3,3}Bitruncated 10-simplex 2475 4959t0,3{3,3,3,3,3,3,3,3,3}Runcinated 10-simplex 15840 132010t1,3{3,3,3,3,3,3,3,3,3}Bicantellated 10-simplex 17820 198011t2,3{3,3,3,3,3,3,3,3,3}Tritruncated 10-simplex 6600 132012t0,4{3,3,3,3,3,3,3,3,3}Stericated 10-simplex 32340 231013t1,4{3,3,3,3,3,3,3,3,3}Biruncinated 10-simplex 55440 462014t2,4{3,3,3,3,3,3,3,3,3}Tricantellated 10-simplex 41580 462015t3,4{3,3,3,3,3,3,3,3,3}Quadritruncated 10-simplex 11550 231016t0,5{3,3,3,3,3,3,3,3,3}Pentellated 10-simplex 41580 277217t1,5{3,3,3,3,3,3,3,3,3}Bistericated 10-simplex 97020 693018t2,5{3,3,3,3,3,3,3,3,3}Triruncinated 10-simplex 110880 924019t3,5{3,3,3,3,3,3,3,3,3}Quadricantellated 10-simplex 62370 693020t4,5{3,3,3,3,3,3,3,3,3}Quintitruncated 10-simplex 13860 277221t0,6{3,3,3,3,3,3,3,3,3}Hexicated 10-simplex 34650 231022t1,6{3,3,3,3,3,3,3,3,3}Bipentellated 10-simplex 103950 693023t2,6{3,3,3,3,3,3,3,3,3}Tristericated 10-simplex 161700 1155024t3,6{3,3,3,3,3,3,3,3,3}Quadriruncinated 10-simplex 138600 1155025t0,7{3,3,3,3,3,3,3,3,3}Heptellated 10-simplex 18480 132026t1,7{3,3,3,3,3,3,3,3,3}Bihexicated 10-simplex 69300 462027t2,7{3,3,3,3,3,3,3,3,3}Tripentellated 10-simplex 138600 924028t0,8{3,3,3,3,3,3,3,3,3}Octellated 10-simplex 5940 49529t1,8{3,3,3,3,3,3,3,3,3}Biheptellated 10-simplex 27720 198030t0,9{3,3,3,3,3,3,3,3,3}Ennecated 10-simplex 990 11031 t0,1,2,3,4,5,6,7,8,9{3,3,3,3,3,3,3,3,3}Omnitruncated 10-simplex 19958400039916800
The B10 family 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.
#GraphCoxeter-Dynkin diagramSchläfli symbolNameElement counts 9-faces 8-faces 7-faces 6-faces 5-faces 4-faces Cells Faces Edges Vertices1 t0{4,3,3,3,3,3,3,3,3}10-cube (deker)2018096033608064134401536011520512010242 t0,1{4,3,3,3,3,3,3,3,3}Truncated 10-cube (tade)51200102403 t1{4,3,3,3,3,3,3,3,3}Rectified 10-cube (rade)4608051204 t2{4,3,3,3,3,3,3,3,3}Birectified 10-cube (brade)184320115205 t3{4,3,3,3,3,3,3,3,3}Trirectified 10-cube (trade)322560153606 t4{4,3,3,3,3,3,3,3,3}Quadrirectified 10-cube (terade)322560134407 t4{3,3,3,3,3,3,3,3,4}Quadrirectified 10-orthoplex (terake)20160080648 t3{3,3,3,3,3,3,3,4}Trirectified 10-orthoplex (trake)8064033609 t2{3,3,3,3,3,3,3,3,4}Birectified 10-orthoplex (brake)2016096010 t1{3,3,3,3,3,3,3,3,4}Rectified 10-orthoplex (rake)288018011 t0,1{3,3,3,3,3,3,3,3,4}Truncated 10-orthoplex (take)306036012 t0{3,3,3,3,3,3,3,3,4}10-orthoplex (ka)102451201152015360134408064336096018020
The D10 family The D10 family has symmetry of order 1,857,945,600 (10 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. 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.
#GraphCoxeter-Dynkin diagramSchläfli symbolNameElement counts 9-faces 8-faces 7-faces 6-faces 5-faces 4-faces Cells Faces Edges Vertices110-demicube (hede)5325300240006480011558414246412288061440115205122Truncated 10-demicube (thede) 195840 23040
Regular and uniform honeycombs There are four fundamental affine Coxeter groups that generate regular and uniform tessellations in 9-space: #Coxeter groupCoxeter-Dynkin diagram1[3]2[4,37,4]3h[4,37,4][4,36,31,1]4q[4,37,4][31,1,35,31,1]
Regular and uniform tessellations include: Regular 9-hypercubic honeycomb, with symbols {4,37,4}, Uniform alternated 9-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. However, there are 3 paracompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.
= [31,1,34,32,1]: = [4,35,32,1]: or = [36,2,1]:
Three honeycombs from the family, generated by end-ringed Coxeter diagrams are: 621 honeycomb: 261 honeycomb: 162 honeycomb:
References
T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
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, 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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, (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: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links Polytope names Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary