Uniform 10-polytope

{{Short description|Type of geometrical object}} {| align=right class=wikitable width=300 style="margin-left:1em;" |+ Graphs of three [[List of regular polytopes#Dimension 5 and higher|regular]] and related [[uniform polytope]]s. |- |- align=center valign=top |colspan=4|[[File:10-simplex t0.svg|100px]]<br /><small>[[10-simplex]]</small> |colspan=4|[[File:10-simplex t01.svg|100px]]<br /><small>[[Truncated 10-simplex]]</small> |colspan=4|[[File:10-simplex t1.svg|100px]]<br /><small>[[Rectified 10-simplex]]</small> |- align=center valign=top |colspan=6|[[File:10-simplex t02.svg|150px]]<br /><small>[[Cantellated 10-simplex]]</small> |colspan=6|[[File:10-simplex t03.svg|150px]]<br /><small>[[Runcinated 10-simplex]]</small> |- align=center valign=top |colspan=4|[[File:10-simplex t04.svg|100px]]<br /><small>[[Stericated 10-simplex]]</small> |colspan=4|[[File:10-simplex t05.svg|100px]]<br /><small>[[Pentellated 10-simplex]]</small> |colspan=4|[[File:10-simplex t06.svg|100px]]<br /><small>[[Hexicated 10-simplex]]</small> |- align=center valign=top |colspan=4|[[File:10-simplex t07.svg|100px]]<br />[[Heptellated 10-simplex]] |colspan=4|[[File:10-simplex t08.svg|100px]]<br />[[Octellated 10-simplex]] |colspan=4|[[File:10-simplex t09.svg|100px]]<br />[[Ennecated 10-simplex]] |- align=center valign=top |colspan=4|[[File:10-orthoplex.svg|100px]]<br />[[10-orthoplex]] |colspan=4|[[File:Truncated 10-orthoplex.png|100px]]<br />[[Truncated 10-orthoplex]] |colspan=4|[[File:Rectified decacross.png|100px]]<br />[[Rectified 10-orthoplex]] |- align=center valign=top |colspan=4|[[File:10-cube.svg|100px]]<br />[[10-cube]] |colspan=4|[[File:Truncated 10-cube.png|100px]]<br />[[Truncated 10-cube]] |colspan=4|[[File:Rectified 10-cube.png|100px]]<br />[[Rectified 10-cube]] |- align=center valign=top |colspan=6|[[File:10-demicube.svg|150px]]<br />[[10-demicube]] |colspan=6|[[File:Truncated 10-demicube.png|150px]]<br />[[Truncated 10-demicube]] |} In ten-dimensional [[geometry]], a 10-polytope is a 10-dimensional [[polytope]] whose boundary consists of [[9-polytope]] [[Facet (mathematics)|facets]], exactly two such facets meeting at each [[8-polytope]] [[Ridge (geometry)|ridge]].

A '''uniform 10-polytope''' is one which is [[vertex-transitive]], and constructed from [[uniform 9-polytope|uniform]] [[Facet (geometry)|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 [[Facet (mathematics)|facets]] around each [[Peak (geometry)|peak]].

There are exactly three such [[List of regular polytopes#Convex 4|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 number]]s and [[torsion coefficient (topology)|torsion coefficient]]s.<ref name="richeson">Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008.</ref>

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.<ref name="richeson"/>

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.<ref name="richeson"/>

== 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 diagram]]s:

{| class=wikitable !# !colspan=2|[[Coxeter group]] ![[Coxeter-Dynkin diagram]] |- |1||A₁₀|| [3⁹]||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} |- |2||B₁₀||[4,3⁸]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} |- |3||D₁₀||[3^7,1,1]||{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} |}

Selected regular and uniform 10-polytopes from each family include: # [[Simplex]] family: A₁₀ [3⁹] - {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} #* 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular: #*# {3⁹} - '''[[10-simplex]]''' - {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} # [[Hypercube]]/[[orthoplex]] family: B₁₀ [4,3⁸] - {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} #* 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones: #*# {4,3⁸} - '''[[10-cube]]''' or '''dekeract''' - {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} #*# {3⁸,4} - '''[[10-orthoplex]]''' or '''decacross''' - {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} #*# h{4,3⁸} - '''[[10-demicube]]''' - {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} # [[Demihypercube]] D₁₀ family: [3^7,1,1] - {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} #* 767 uniform 10-polytopes as permutations of rings in the group diagram, including: #*# '''1_7,1''' - '''[[10-demicube]]''' or '''demidekeract''' - {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} #*# '''7_1,1''' - '''[[10-orthoplex]]''' - {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}

== 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 diagram]]s 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.

{| class="wikitable" !rowspan=2|# !rowspan=2|Graph !rowspan=2|[[Coxeter-Dynkin diagram]]<br />[[Schläfli symbol]]<br />Name !colspan=10|Element counts |- || 9-faces|| 8-faces|| 7-faces|| 6-faces|| 5-faces|| 4-faces|| Cells|| Faces|| Edges|| Vertices |- |- align=center !1 |[[File:10-simplex t0.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t₀{3,3,3,3,3,3,3,3,3}<br />[[10-simplex]] (ux) |11||55||165||330||462||462||330||165||55||11 |- align=center !2 |[[File:10-simplex t1.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t₁{3,3,3,3,3,3,3,3,3}<br />[[Rectified 10-simplex]] (ru) || || || || || || || || ||495 ||55 |- align=center !3 |[[File:10-simplex t2.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}<br />t₂{3,3,3,3,3,3,3,3,3}<br />[[Birectified 10-simplex]] (bru) || || || || || || || || ||1980 ||165 |- align=center !4 |[[File:10-simplex t3.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}<br />t₃{3,3,3,3,3,3,3,3,3}<br />[[Trirectified 10-simplex]] (tru) || || || || || || || || ||4620 ||330 |- align=center !5 |[[File:10-simplex t4.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}<br />t₄{3,3,3,3,3,3,3,3,3}<br />[[Quadrirectified 10-simplex]] (teru) || || || || || || || || ||6930 ||462 |- align=center !6 |[[File:10-simplex t01.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}<br />t_0,1{3,3,3,3,3,3,3,3,3}<br />[[Truncated 10-simplex]] (tu) || || || || || || || || ||550 ||110 |- align=center !7 |[[File:10-simplex t02.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}<br />t_0,2{3,3,3,3,3,3,3,3,3}<br />[[Cantellated 10-simplex]] || || || || || || || || ||4455 ||495 |- align=center !8 |[[File:10-simplex t12.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node}}<br />t_1,2{3,3,3,3,3,3,3,3,3}<br />[[Bitruncated 10-simplex]] || || || || || || || || ||2475 ||495 |- align=center !9 |[[File:10-simplex t03.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}<br />t_0,3{3,3,3,3,3,3,3,3,3}<br />[[Runcinated 10-simplex]] || || || || || || || || ||15840 ||1320 |- align=center !10 |[[File:10-simplex t13.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node}}<br />t_1,3{3,3,3,3,3,3,3,3,3}<br />[[Bicantellated 10-simplex]] || || || || || || || || ||17820 ||1980 |- align=center !11 |[[File:10-simplex t23.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node}}<br />t_2,3{3,3,3,3,3,3,3,3,3}<br />[[Tritruncated 10-simplex]] || || || || || || || || ||6600 ||1320 |- align=center !12 |[[File:10-simplex t04.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1}}<br />t_0,4{3,3,3,3,3,3,3,3,3}<br />[[Stericated 10-simplex]] || || || || || || || || ||32340 ||2310 |- align=center !13 |[[File:10-simplex t14.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node}}<br />t_1,4{3,3,3,3,3,3,3,3,3}<br />[[Biruncinated 10-simplex]] || || || || || || || || ||55440 ||4620 |- align=center !14 |[[File:10-simplex t24.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node}}<br />t_2,4{3,3,3,3,3,3,3,3,3}<br />[[Tricantellated 10-simplex]] || || || || || || || || ||41580 ||4620 |- align=center !15 | | {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}<br />t_3,4{3,3,3,3,3,3,3,3,3}<br />[[Quadritruncated 10-simplex]] || || || || || || || || ||11550 ||2310 |- align=center !16 |[[File:10-simplex t05.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1}}<br />t_0,5{3,3,3,3,3,3,3,3,3}<br />[[Pentellated 10-simplex]] || || || || || || || || ||41580 ||2772 |- align=center !17 | | {{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node}}<br />t_1,5{3,3,3,3,3,3,3,3,3}<br />[[Bistericated 10-simplex]] || || || || || || || || ||97020 ||6930 |- align=center !18 | | {{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node}}<br />t_2,5{3,3,3,3,3,3,3,3,3}<br />[[Triruncinated 10-simplex]] || || || || || || || || ||110880 ||9240 |- align=center !19 |[[File:10-simplex t35.svg|60px]] | {{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}<br />t_3,5{3,3,3,3,3,3,3,3,3}<br />[[Quadricantellated 10-simplex]] || || || || || || || || ||62370 ||6930 |- align=center BGCOLOR="#e0f0e0" !20 | | {{CDD|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}<br />t_4,5{3,3,3,3,3,3,3,3,3}<br />[[Quintitruncated 10-simplex]] || || || || || || || || ||13860 ||2772 |- align=center !21 |[[File:10-simplex t06.svg|60px]] | {{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t_0,6{3,3,3,3,3,3,3,3,3}<br />[[Hexicated 10-simplex]] || || || || || || || || ||34650 ||2310 |- align=center !22 | | {{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t_1,6{3,3,3,3,3,3,3,3,3}<br />[[Bipentellated 10-simplex]] || || || || || || || || ||103950 ||6930 |- align=center !23 | | {{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node|3|node}}<br />t_2,6{3,3,3,3,3,3,3,3,3}<br />[[Tristericated 10-simplex]] || || || || || || || || ||161700 ||11550 |- align=center BGCOLOR="#e0f0e0" !24 | | {{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node}}<br />t_3,6{3,3,3,3,3,3,3,3,3}<br />[[Quadriruncinated 10-simplex]] || || || || || || || || ||138600 ||11550 |- align=center !25 |[[File:10-simplex t07.svg|60px]] | {{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t_0,7{3,3,3,3,3,3,3,3,3}<br />[[Heptellated 10-simplex]] || || || || || || || || ||18480 ||1320 |- align=center !26 | | {{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t_1,7{3,3,3,3,3,3,3,3,3}<br />[[Bihexicated 10-simplex]] || || || || || || || || ||69300 ||4620 |- align=center BGCOLOR="#e0f0e0" !27 | | {{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}<br />t_2,7{3,3,3,3,3,3,3,3,3}<br />[[Tripentellated 10-simplex]] || || || || || || || || ||138600 ||9240 |- align=center !28 |[[File:10-simplex t08.svg|60px]] | {{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t_0,8{3,3,3,3,3,3,3,3,3}<br />[[Octellated 10-simplex]] || || || || || || || || ||5940 ||495 |- align=center BGCOLOR="#e0f0e0" !29 | | {{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t_1,8{3,3,3,3,3,3,3,3,3}<br />[[Biheptellated 10-simplex]] || || || || || || || || ||27720 ||1980 |- align=center BGCOLOR="#e0f0e0" !30 |[[File:10-simplex t09.svg|60px]] | {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t_0,9{3,3,3,3,3,3,3,3,3}<br />[[Ennecated 10-simplex]] || || || || || || || || ||990 ||110 |- align=center !31 | | {{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<br />t_{0,1,2,3,4,5,6,7,8,9}{3,3,3,3,3,3,3,3,3}<br />[[Omnitruncated 10-simplex]] || || || || || || || || ||199584000||39916800 |}

== The B₁₀ family ==

There are 1023 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings.

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

{| class="wikitable" !rowspan=2|# !rowspan=2|Graph !rowspan=2|[[Coxeter-Dynkin diagram]]<br />[[Schläfli symbol]]<br />Name !colspan=10|Element counts |- ! 9-faces ! 8-faces ! 7-faces ! 6-faces ! 5-faces ! 4-faces ! Cells ! Faces ! Edges ! Vertices |- align=center !1 |[[File:10-cube t0.svg|60px]] | {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t₀{4,3,3,3,3,3,3,3,3}<br />[[10-cube]] (deker) |20||180||960||3360||8064||13440||15360||11520||5120||1024 |- align=center !2 |[[File:Truncated 10-cube.png|60px]] | {{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t_0,1{4,3,3,3,3,3,3,3,3}<br />[[Truncated 10-cube]] (tade) | | | | | | | | |51200 |10240 |- align=center !3 |[[File:10-cube t1.svg|60px]] | {{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t₁{4,3,3,3,3,3,3,3,3}<br />[[Rectified 10-cube]] (rade) | | | | | | | | |46080 |5120 |- align=center !4 |[[File:10-cube t2.svg|60px]] | {{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t₂{4,3,3,3,3,3,3,3,3}<br />[[Birectified 10-cube]] (brade) | | | | | | | | |184320 |11520 |- align=center !5 |[[File:10-cube t3.svg|60px]] | {{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t₃{4,3,3,3,3,3,3,3,3}<br />[[Trirectified 10-cube]] (trade) | | | | | | | | |322560 |15360 |- align=center !6 |[[File:10-cube t4.svg|60px]] | {{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}<br />t₄{4,3,3,3,3,3,3,3,3}<br />[[Quadrirectified 10-cube]] (terade) | | | | | | | | |322560 |13440 |- align=center !7 |[[File:10-cube t5.svg|60px]] | {{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}<br />t₄{3,3,3,3,3,3,3,3,4}<br />[[Quadrirectified 10-orthoplex]] (terake) | | | | | | | | |201600 |8064 |- align=center !8 |[[File:10-cube t6.svg|60px]] | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}<br />t₃{3,3,3,3,3,3,3,4}<br />[[Trirectified 10-orthoplex]] (trake) | | | | | | | | |80640 |3360 |- align=center !9 |[[File:10-cube t7.svg|60px]] | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}<br />t₂{3,3,3,3,3,3,3,3,4}<br />[[Birectified 10-orthoplex]] (brake) | | | | | | | | |20160 |960 |- align=center !10 |[[File:10-cube t8.svg|60px]] | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t₁{3,3,3,3,3,3,3,3,4}<br />[[Rectified 10-orthoplex]] (rake) | | | | | | | | |2880 |180 |- align=center !11 |[[File:Truncated 10-orthoplex.png|60px]] | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}<br />t_0,1{3,3,3,3,3,3,3,3,4}<br />[[Truncated 10-orthoplex]] (take) | | | | | | | | |3060 |360 |- align=center !12 |[[File:10-cube t9.svg|60px]] | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t₀{3,3,3,3,3,3,3,3,4}<br />[[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]] × 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.

{| class="wikitable" !rowspan=2|# !rowspan=2|Graph !rowspan=2|[[Coxeter-Dynkin diagram]]<br />[[Schläfli symbol]]<br />Name !colspan=10|Element counts |- ! 9-faces ! 8-faces ! 7-faces ! 6-faces ! 5-faces ! 4-faces ! Cells ! Faces ! Edges ! Vertices |- align=center |1||[[File:10-demicube.svg|60px]]||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />[[10-demicube]] (hede) |532||5300||24000||64800||115584||142464||122880||61440||11520||512 |- align=center |2||[[File:Truncated 10-demicube.png|60px]]||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />[[Truncated 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: {| class=wikitable !# !colspan=2|[[Coxeter group]] ![[Coxeter-Dynkin diagram]] |- |1||<math>{\tilde{A}}_9</math>||[3^[10]]||{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}} |- |2||<math>{\tilde{B}}_9</math>||[4,3⁷,4]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} |- |3||<math>{\tilde{C}}_9</math>||h[4,3⁷,4]<br />[4,3⁶,3^1,1]||{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} |- |4||<math>{\tilde{D}}_9</math>||q[4,3⁷,4]<br />[3^1,1,3⁵,3^1,1]||{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}} |}

Regular and uniform tessellations include: * [[List of regular polytopes#Higher dimensions 3|Regular]] [[Hypercubic honeycomb|9-hypercubic honeycomb]], with symbols {4,3⁷,4}, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} * Uniform [[Alternated hypercubic honeycomb|alternated 9-hypercubic honeycomb]] with symbols h{4,3⁷,4}, {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}

=== 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 [[Coxeter-Dynkin diagram#Rank 4 to 10|3 paracompact hyperbolic Coxeter groups]] of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.

{| class=wikitable |align=right|<math>{\bar{Q}}_9</math> = [3^1,1,3⁴,3^2,1]:<BR>{{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}} |align=right|<math>{\bar{S}}_9</math> = [4,3⁵,3^2,1]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|4a|nodea}} |align=right|<math>E_{10}</math> or <math>{\bar{T}}_9</math> = [3^6,2,1]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}} |}

Three honeycombs from the <math>E_{10}</math> family, generated by end-ringed Coxeter diagrams are: * [[6 21 honeycomb|6₂₁ honeycomb]]: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}} * [[2 61 honeycomb|2₆₁ honeycomb]]: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}} * [[1 62 honeycomb|1₆₂ honeycomb]]: {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

== References == {{reflist}} * [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900 * {{cite journal|year=1910|author=A. Boole Stott|authorlink=Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|journal=Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam.|volume=XI|number=1|publisher=Johannes Müller|location=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf|archive-url=https://web.archive.org/web/20250429000816/https://dwc.knaw.nl/DL/publications/PU00011492.pdf|archive-date=29 April 2025}} * [[Harold Scott MacDonald Coxeter|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 * '''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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] {{Webarchive|url=https://web.archive.org/web/20160711140441/http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html |date=2016-07-11 }} ** (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] * [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 * {{KlitzingPolytopes|polyxenna.htm|10D|uniform polytopes (polyxenna)}}

== External links == * [http://www.steelpillow.com/polyhedra/ditela.html Polytope names] * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary] * {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}} {{Polytopes}}

[[Category:10-polytopes]]