Uniform 7-polytope

{{Short description|Seven-dimensional geometric object}} {| class="wikitable skin-invert-image" style="float:right; margin-left:1em; width:300px" |+ Graphs of three [[List of regular polytopes#Convex 4|regular]] and related [[uniform polytope]]s |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:7-simplex t0.svg|100px]]<br/>[[7-simplex]] |colspan=4|[[File:7-simplex t1.svg|100px]]<br/>[[Rectified 7-simplex]] |colspan=4|[[File:7-simplex t01.svg|100px]]<br/>[[Truncated 7-simplex]] |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:7-simplex t02.svg|100px]]<br/>[[Cantellated 7-simplex]] |colspan=4|[[File:7-simplex t03.svg|100px]]<br/>[[Runcinated 7-simplex]] |colspan=4|[[File:7-simplex t04.svg|100px]]<br/>[[Stericated 7-simplex]] |- style="vertical-align:top; text-align:center;" |colspan=6|[[File:7-simplex t05.svg|150px]]<br/>Pentellated 7-simplex |colspan=6|[[File:7-simplex t06.svg|150px]]<br/>[[Hexicated 7-simplex]] |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:7-cube t6.svg|100px]]<br/>[[7-orthoplex]] |colspan=4|[[File:7-cube t56.svg|100px]]<br/>[[Truncated 7-orthoplex]] |colspan=4|[[File:7-cube t5.svg|100px]]<br/>[[Rectified 7-orthoplex]] |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:7-cube t46.svg|100px]]<br/>[[Cantellated 7-orthoplex]] |colspan=4|[[File:7-cube t36.svg|100px]]<br/>[[Runcinated 7-orthoplex]] |colspan=4|[[File:7-cube t26.svg|100px]]<br/>[[Stericated 7-orthoplex]] |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:7-cube t16.svg|100px]]<br/>[[Pentellated 7-orthoplex]] |colspan=4|[[File:7-cube t06.svg|100px]]<br/>Hexicated 7-cube |colspan=4|[[File:7-cube t05.svg|100px]]<br/>[[Pentellated 7-cube]] |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:7-cube t04.svg|100px]]<br/>[[Stericated 7-cube]] |colspan=4|[[File:7-cube t02.svg|100px]]<br/>[[Cantellated 7-cube]] |colspan=4|[[File:7-cube t03.svg|100px]]<br/>[[Runcinated 7-cube]] |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:7-cube t0.svg|100px]]<br/>[[7-cube]] |colspan=4|[[File:7-cube t01.svg|100px]]<br/>[[Truncated 7-cube]] |colspan=4|[[File:7-cube t1.svg|100px]]<br/>[[Rectified 7-cube]] |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:7-demicube t0 D7.svg|100px]]<br/>[[7-demicube]] |colspan=4|[[File:7-demicube t01 D7.svg|100px]]<br/>[[Cantic 7-cube]] |colspan=4|[[File:7-demicube t02 D7.svg|100px]]<br/>[[Runcic 7-cube]] |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:7-demicube t03 D7.svg|100px]]<br/>[[Steric 7-cube]] |colspan=4|[[File:7-demicube t04 D7.svg|100px]]<br/>[[Pentic 7-cube]] |colspan=4|[[File:7-demicube t05 D7.svg|100px]]<br/>[[Hexic 7-cube]] |- style="vertical-align:top; text-align:center;" |colspan=4|[[File:E7 graph.svg|100px]]<br/>[[3 21 polytope|3₂₁]] |colspan=4|[[File:Gosset 2 31 polytope.svg|100px]]<br/>[[2 31 polytope|2₃₁]] |colspan=4|[[File:Gosset 1 32 petrie.svg|100px]]<br/>[[1 32 polytope|1₃₂]] |} In [[seven-dimensional space|seven-dimensional]] [[geometry]], a '''7-polytope''' is a [[polytope]] contained by 6-polytope facets. Each [[5-polytope]] [[Ridge (geometry)|ridge]] being shared by exactly two [[6-polytope]] [[Facet (mathematics)|facets]].

A '''uniform 7-polytope''' is one whose symmetry group is [[vertex-transitive|transitive on vertices]] and whose facets are [[uniform 6-polytope]]s.

== Regular 7-polytopes == Regular 7-polytopes are represented by the [[Schläfli symbol]] {p,q,r,s,t,u} with '''u''' {p,q,r,s,t} 6-polytopes [[Facet (mathematics)|facets]] around each 4-face.

There are exactly three such [[List of regular polytopes#Convex 4|convex regular 7-polytopes]]: # {3,3,3,3,3,3} - [[7-simplex]] # {4,3,3,3,3,3} - [[7-cube]] # {3,3,3,3,3,4} - [[7-orthoplex]]

There are no nonconvex regular 7-polytopes.

== Characteristics == The topology of any given 7-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, 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 7-polytopes by fundamental Coxeter groups == Uniform 7-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the [[Coxeter-Dynkin diagram]]s:

{| class="wikitable" |- !# !colspan=3|[[Coxeter group]] !Regular and semiregular forms !Uniform count |- |1||A₇|| [3⁶]||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node}} | * [[7-simplex]] - {3⁶}, {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}} |71 |- |2||B₇||[4,3⁵]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node}} | * [[7-cube]] - {4,3⁵}, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}} * [[7-orthoplex]] - {3⁵,4}, {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|4|node}} * [[7-demicube]] - h{4,3⁵}, {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node}} |127 + 32 |- |3||D₇||[3^3,1,1]||{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node}} | * [[7-demicube]], {3,3^4,1}, {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}} * [[7-orthoplex]], {3⁴,3^1,1}, {{CDD|node_1|3|node|3|node|3|node|3|node|split1|nodes}} |95 (0 unique) |- |4||[[E7 (mathematics)|E₇]]||[3^3,2,1]||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}} | * '''[[Gosset 3 21 polytope|3₂₁]]''' - {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}} * '''[[Gosset 1 32 polytope|1₃₂]]''' - {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}} * '''[[Gosset 2 31 polytope|2₃₁]]''' - {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}} |127 |}

{| class="wikitable collapsed collapsible" !colspan=12|Prismatic finite Coxeter groups |- !# !colspan=2|[[Coxeter group]] ![[Coxeter diagram]] |- !colspan=4|6+1 |- |1 ||A₆A₁|| [3⁵]×[&nbsp;]|| {{CDD|node|3|node|3|node|3|node|3|node|3|node|2|node}} |- |2 ||BC₆A₁|| [4,3⁴]×[&nbsp;]|| {{CDD|node|4|node|3|node|3|node|3|node|3|node|2|node}} |- |3 ||D₆A₁|| [3^3,1,1]×[&nbsp;]|| {{CDD|nodes|split2|node|3|node|3|node|3|node|2|node}} |- |4 ||E₆A₁|| [3^2,2,1]×[&nbsp;]|| {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|2|nodea}} |- !colspan=4|5+2 |- |1 ||A₅I₂(p)|| [3,3,3]×[p]|| {{CDD|node|3|node|3|node|3|node|3|node|2|node|p|node}} |- |2 ||BC₅I₂(p)|| [4,3,3]×[p]|| {{CDD|node|4|node|3|node|3|node|3|node|2|node|p|node}} |- |3 ||D₅I₂(p)|| [3^2,1,1]×[p]|| {{CDD|nodes|split2|node|3|node|3|node|2|node|p|node}} |- !colspan=4|5+1+1 |- |1 ||A₅A₁²|| [3,3,3]×[&nbsp;]²|| {{CDD|node|3|node|3|node|3|node|3|node|2|node|2|node}} |- |2 ||BC₅A₁²|| [4,3,3]×[&nbsp;]²|| {{CDD|node|4|node|3|node|3|node|3|node|2|node|2|node}} |- |3 ||D₅A₁²|| [3^2,1,1]×[&nbsp;]²|| {{CDD|nodes|split2|node|3|node|3|node|2|node|2|node}} |- !colspan=4|4+3 |- |1 ||A₄A₃|| [3,3,3]×[3,3]|| {{CDD|node|3|node|3|node|3|node|2|node|3|node|3|node}} |- |2 ||A₄B₃|| [3,3,3]×[4,3]|| {{CDD|node|3|node|3|node|3|node|2|node|4|node|3|node}} |- |3 ||A₄H₃|| [3,3,3]×[5,3]|| {{CDD|node|3|node|3|node|3|node|2|node|5|node|3|node}} |- |4 ||BC₄A₃|| [4,3,3]×[3,3]|| {{CDD|node|4|node|3|node|3|node|2|node|3|node|3|node}} |- |5 ||BC₄B₃|| [4,3,3]×[4,3]|| {{CDD|node|4|node|3|node|3|node|2|node|4|node|3|node}} |- |6 ||BC₄H₃|| [4,3,3]×[5,3]|| {{CDD|node|4|node|3|node|3|node|2|node|5|node|3|node}} |- |7 ||H₄A₃|| [5,3,3]×[3,3]|| {{CDD|node|5|node|3|node|3|node|2|node|3|node|3|node}} |- |8 ||H₄B₃|| [5,3,3]×[4,3]|| {{CDD|node|5|node|3|node|3|node|2|node|4|node|3|node}} |- |9 ||H₄H₃|| [5,3,3]×[5,3]|| {{CDD|node|5|node|3|node|3|node|2|node|5|node|3|node}} |- |10 ||F₄A₃|| [3,4,3]×[3,3]|| {{CDD|node|3|node|4|node|3|node|2|node|3|node|3|node}} |- |11 ||F₄B₃|| [3,4,3]×[4,3]|| {{CDD|node|3|node|4|node|3|node|2|node|4|node|3|node}} |- |12 ||F₄H₃|| [3,4,3]×[5,3]|| {{CDD|node|3|node|4|node|3|node|2|node|5|node|3|node}} |- |13 ||D₄A₃|| [3^1,1,1]×[3,3]|| {{CDD|nodes|split2|node|3|node|2|node|3|node|3|node}} |- |14 ||D₄B₃|| [3^1,1,1]×[4,3]|| {{CDD|nodes|split2|node|3|node|2|node|4|node|3|node}} |- |15 ||D₄H₃|| [3^1,1,1]×[5,3]|| {{CDD|nodes|split2|node|3|node|2|node|5|node|3|node}} |- !colspan=4|4+2+1 |- |1 ||A₄I₂(p)A₁|| [3,3,3]×[p]×[&nbsp;]|| {{CDD|node|3|node|3|node|3|node|2|node|p|node|2|node}} |- |2 ||BC₄I₂(p)A₁|| [4,3,3]×[p]×[&nbsp;]|| {{CDD|node|4|node|3|node|3|node|2|node|p|node|2|node}} |- |3 ||F₄I₂(p)A₁|| [3,4,3]×[p]×[&nbsp;]|| {{CDD|node|3|node|4|node|3|node|2|node|p|node|2|node}} |- |4 ||H₄I₂(p)A₁|| [5,3,3]×[p]×[&nbsp;]|| {{CDD|node|5|node|3|node|3|node|2|node|p|node|2|node}} |- |5 ||D₄I₂(p)A₁|| [3^1,1,1]×[p]×[&nbsp;]|| {{CDD|nodes|split2|node|3|node|2|node|p|node|2|node}} |- !colspan=4|4+1+1+1 |- |1 ||A₄A₁³|| [3,3,3]×[&nbsp;]³|| {{CDD|node|3|node|3|node|3|node|2|node|2|node|2|node}} |- |2 ||BC₄A₁³|| [4,3,3]×[&nbsp;]³|| {{CDD|node|4|node|3|node|3|node|2|node|2|node|2|node}} |- |3 ||F₄A₁³|| [3,4,3]×[&nbsp;]³|| {{CDD|node|3|node|4|node|3|node|2|node|2|node|2|node}} |- |4 ||H₄A₁³|| [5,3,3]×[&nbsp;]³|| {{CDD|node|5|node|3|node|3|node|2|node|2|node|2|node}} |- |5 ||D₄A₁³|| [3^1,1,1]×[&nbsp;]³|| {{CDD|nodes|split2|node|3|node|2|node|2|node|2|node}} |- !colspan=4|3+3+1 |- |1 ||A₃A₃A₁|| [3,3]×[3,3]×[&nbsp;]|| {{CDD|node|3|node|3|node|2|node|3|node|3|node|2|node}} |- |2 ||A₃B₃A₁|| [3,3]×[4,3]×[&nbsp;]|| {{CDD|node|3|node|3|node|2|node|4|node|3|node|2|node}} |- |3 ||A₃H₃A₁|| [3,3]×[5,3]×[&nbsp;]|| {{CDD|node|3|node|3|node|2|node|5|node|3|node|2|node}} |- |4 ||BC₃B₃A₁|| [4,3]×[4,3]×[&nbsp;]|| {{CDD|node|4|node|3|node|2|node|4|node|3|node|2|node}} |- |5 ||BC₃H₃A₁|| [4,3]×[5,3]×[&nbsp;]|| {{CDD|node|4|node|3|node|2|node|5|node|3|node|2|node}} |- |6 ||H₃A₃A₁|| [5,3]×[5,3]×[&nbsp;]|| {{CDD|node|5|node|3|node|2|node|5|node|3|node|2|node}} |- !colspan=4|3+2+2 |- |1 ||A₃I₂(p)I₂(q)|| [3,3]×[p]×[q]|| {{CDD|node|3|node|3|node|2|node|p|node|2|node|q|node}} |- |2 ||BC₃I₂(p)I₂(q)|| [4,3]×[p]×[q]|| {{CDD|node|4|node|3|node|2|node|p|node|2|node|q|node}} |- |3 ||H₃I₂(p)I₂(q)|| [5,3]×[p]×[q]|| {{CDD|node|5|node|3|node|2|node|p|node|2|node|q|node}} |- !colspan=4|3+2+1+1 |- |1 ||A₃I₂(p)A₁²|| [3,3]×[p]×[&nbsp;]²|| {{CDD|node|3|node|3|node|2|node|p|node|2|node|2|node}} |- |2 ||BC₃I₂(p)A₁²|| [4,3]×[p]×[&nbsp;]²|| {{CDD|node|4|node|3|node|2|node|p|node|2|node|2|node}} |- |3 ||H₃I₂(p)A₁²|| [5,3]×[p]×[&nbsp;]²|| {{CDD|node|5|node|3|node|2|node|p|node|2|node|2|node}} |- !colspan=4|3+1+1+1+1 |- |1 ||A₃A₁⁴|| [3,3]×[&nbsp;]⁴|| {{CDD|node|3|node|3|node|2|node|2|node|2|node|2|node}} |- |2 ||BC₃A₁⁴|| [4,3]×[&nbsp;]⁴|| {{CDD|node|4|node|3|node|2|node|2|node|2|node|2|node}} |- |3 ||H₃A₁⁴|| [5,3]×[&nbsp;]⁴|| {{CDD|node|5|node|3|node|2|node|2|node|2|node|2|node}} |- !colspan=4|2+2+2+1 |- |1 ||I₂(p)I₂(q)I₂(r)A₁|| [p]×[q]×[r]×[&nbsp;]|| {{CDD|node|p|node|2|node|q|node|2|node|r|node|2|node}} |- !colspan=4|2+2+1+1+1 |- |1 ||I₂(p)I₂(q)A₁³|| [p]×[q]×[&nbsp;]³|| {{CDD|node|p|node|2|node|q|node|2|node|2|node|2|node}} |- !colspan=4|2+1+1+1+1+1 |- |1 ||I₂(p)A₁⁵|| [p]×[&nbsp;]⁵|| {{CDD|node|p|node|2|node|2|node|2|node|2|node|2|node}} |- !colspan=4|1+1+1+1+1+1+1 |- |1 ||A₁⁷|| [&nbsp;]⁷|| {{CDD|node|2|node|2|node|2|node|2|node|2|node|2|node}} |}

== The A₇ family == The A₇ family has symmetry of order 40320 (8 [[factorial]]).

There are 71 (64 + 8 − 1) forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. All 71 are enumerated below. [[Norman Johnson (mathematician)|Norman Johnson]]'s truncation names are given. Bowers names and acronym are also given for cross-referencing.

See also a [[list of A7 polytopes]] for symmetric [[Coxeter plane]] graphs of these polytopes.

{| class="wikitable collapsible collapsed" !colspan=12|A₇ uniform polytopes |- !rowspan=2|# !rowspan=2|[[Coxeter-Dynkin diagram]] !rowspan=2|Truncation<br />indices !rowspan=2|[[Norman Johnson (mathematician)|Johnson name]]<br />Bowers name (and acronym) !rowspan=2|Basepoint !colspan=7|Element counts |- ! 6|| 5|| 4|| 3|| 2|| 1|| 0 |- style="text-align:center;" |1||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}||t₀||[[7-simplex]] (oca)||(0,0,0,0,0,0,0,1)||8||28||56||70||56||28||8 |- style="text-align:center;" |2||{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}||t₁||[[Rectified 7-simplex]] (roc)||(0,0,0,0,0,0,1,1)||16||84||224||350||336||168||28 |- style="text-align:center;" |3||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}||t₂||[[Birectified 7-simplex]] (broc)||(0,0,0,0,0,1,1,1)||16||112||392||770||840||420||56 |- style="text-align:center; background:#e0f0e0;" |4||{{dark mode invert|{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}}}||t₃||[[Trirectified 7-simplex]] (he)||(0,0,0,0,1,1,1,1)||16||112||448||980||1120||560||70 |- style="text-align:center;" |5||{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}||t_0,1||[[Truncated 7-simplex]] (toc)||(0,0,0,0,0,0,1,2)||16||84||224||350||336||196||56 |- style="text-align:center;" |6||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}||t_0,2||[[Cantellated 7-simplex]] (saro)||(0,0,0,0,0,1,1,2)||44||308||980||1750||1876||1008||168 |- style="text-align:center;" |7||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node}}||t_1,2||[[Bitruncated 7-simplex]] (bittoc)||(0,0,0,0,0,1,2,2)||||||||||||588||168 |- style="text-align:center;" |8||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}||t_0,3||[[Runcinated 7-simplex]] (spo)||(0,0,0,0,1,1,1,2)||100||756||2548||4830||4760||2100||280 |- style="text-align:center;" |9||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node}}||t_1,3||[[Bicantellated 7-simplex]] (sabro)||(0,0,0,0,1,1,2,2)||||||||||||2520||420 |- style="text-align:center;" |10||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node}}||t_2,3||[[Tritruncated 7-simplex]] (tattoc)||(0,0,0,0,1,2,2,2)||||||||||||980||280 |- style="text-align:center;" |11||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1}}||t_0,4||[[Stericated 7-simplex]] (sco)||(0,0,0,1,1,1,1,2)||||||||||||2240||280 |- style="text-align:center;" |12||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node}}||t_1,4||[[Biruncinated 7-simplex]] (sibpo)||(0,0,0,1,1,1,2,2)||||||||||||4200||560 |- style="text-align:center; background:#e0f0e0;" |13||{{dark mode invert|{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node}}}}||t_2,4||[[Tricantellated 7-simplex]] (stiroh)||(0,0,0,1,1,2,2,2)||||||||||||3360||560 |- style="text-align:center;" |14||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1}}||t_0,5||Pentellated 7-simplex (seto)||(0,0,1,1,1,1,1,2)||||||||||||1260||168 |- style="text-align:center; background:#e0f0e0;" |15||{{dark mode invert|{{CDD|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node}}}}||t_1,5||[[Bistericated 7-simplex]] (sabach)||(0,0,1,1,1,1,2,2)||||||||||||3360||420 |- style="text-align:center; background:#e0f0e0;" |16||{{dark mode invert|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}}}||t_0,6||[[Hexicated 7-simplex]] (suph)||(0,1,1,1,1,1,1,2)||||||||||||336||56 |- style="text-align:center;" |17||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t_0,1,2||[[Cantitruncated 7-simplex]] (garo)||(0,0,0,0,0,1,2,3)||||||||||||1176||336 |- style="text-align:center;" |18||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t_0,1,3||[[Runcitruncated 7-simplex]] (patto)||(0,0,0,0,1,1,2,3)||||||||||||4620||840 |- style="text-align:center;" |19||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}||t_0,2,3||[[Runcicantellated 7-simplex]] (paro)||(0,0,0,0,1,2,2,3)||||||||||||3360||840 |- style="text-align:center;" |20||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}||t_1,2,3||[[Bicantitruncated 7-simplex]] (gabro)||(0,0,0,0,1,2,3,3)||||||||||||2940||840 |- style="text-align:center;" |21||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}||t_0,1,4||[[Steritruncated 7-simplex]] (cato)||(0,0,0,1,1,1,2,3)||||||||||||7280||1120 |- style="text-align:center;" |22||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}||t_0,2,4||[[Stericantellated 7-simplex]] (caro)||(0,0,0,1,1,2,2,3)||||||||||||10080||1680 |- style="text-align:center;" |23||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}||t_1,2,4||[[Biruncitruncated 7-simplex]] (bipto)||(0,0,0,1,1,2,3,3)||||||||||||8400||1680 |- style="text-align:center;" |24||{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}||t_0,3,4||[[Steriruncinated 7-simplex]] (cepo)||(0,0,0,1,2,2,2,3)||||||||||||5040||1120 |- style="text-align:center;" |25||{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}||t_1,3,4||[[Biruncicantellated 7-simplex]] (bipro)||(0,0,0,1,2,2,3,3)||||||||||||7560||1680 |- style="text-align:center; background:#e0f0e0;" |26||{{dark mode invert|{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}}}||t_2,3,4||[[Tricantitruncated 7-simplex]] (gatroh)||(0,0,0,1,2,3,3,3)||||||||||||3920||1120 |- style="text-align:center;" |27||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}||t_0,1,5||Pentitruncated 7-simplex (teto)||(0,0,1,1,1,1,2,3)||||||||||||5460||840 |- style="text-align:center;" |28||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}||t_0,2,5||Penticantellated 7-simplex (tero)||(0,0,1,1,1,2,2,3)||||||||||||11760||1680 |- style="text-align:center;" |29||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}||t_1,2,5||[[Bisteritruncated 7-simplex]] (bacto)||(0,0,1,1,1,2,3,3)||||||||||||9240||1680 |- style="text-align:center;" |30||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}||t_0,3,5||Pentiruncinated 7-simplex (tepo)||(0,0,1,1,2,2,2,3)||||||||||||10920||1680 |- style="text-align:center; background:#e0f0e0;" |31||{{dark mode invert|{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}}}||t_1,3,5||[[Bistericantellated 7-simplex]] (bacroh)||(0,0,1,1,2,2,3,3)||||||||||||15120||2520 |- style="text-align:center;" |32||{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}||t_0,4,5||Pentistericated 7-simplex (teco)||(0,0,1,2,2,2,2,3)||||||||||||4200||840 |- style="text-align:center;" |33||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}||t_0,1,6||[[Hexitruncated 7-simplex]] (puto)||(0,1,1,1,1,1,2,3)||||||||||||1848||336 |- style="text-align:center;" |34||{{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}||t_0,2,6||[[Hexicantellated 7-simplex]] (puro)||(0,1,1,1,1,2,2,3)||||||||||||5880||840 |- style="text-align:center; background:#e0f0e0;" |35||{{dark mode invert|{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}}}||t_0,3,6||[[Hexiruncinated 7-simplex]] (puph)||(0,1,1,1,2,2,2,3)||||||||||||8400||1120 |- style="text-align:center;" |36||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t_0,1,2,3||[[Runcicantitruncated 7-simplex]] (gapo)||(0,0,0,0,1,2,3,4)||||||||||||5880||1680 |- style="text-align:center;" |37||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t_0,1,2,4||[[Stericantitruncated 7-simplex]] (cagro)||(0,0,0,1,1,2,3,4)||||||||||||16800||3360 |- style="text-align:center;" |38||{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t_0,1,3,4||[[Steriruncitruncated 7-simplex]] (capto)||(0,0,0,1,2,2,3,4)||||||||||||13440||3360 |- style="text-align:center;" |39||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t_0,2,3,4||[[Steriruncicantellated 7-simplex]] (capro)||(0,0,0,1,2,3,3,4)||||||||||||13440||3360 |- style="text-align:center;" |40||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}||t_1,2,3,4||[[Biruncicantitruncated 7-simplex]] (gibpo)||(0,0,0,1,2,3,4,4)||||||||||||11760||3360 |- style="text-align:center;" |41||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t_0,1,2,5||Penticantitruncated 7-simplex (tegro)||(0,0,1,1,1,2,3,4)||||||||||||18480||3360 |- style="text-align:center;" |42||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t_0,1,3,5||Pentiruncitruncated 7-simplex (tapto)||(0,0,1,1,2,2,3,4)||||||||||||27720||5040 |- style="text-align:center;" |43||{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}||t_0,2,3,5||Pentiruncicantellated 7-simplex (tapro)||(0,0,1,1,2,3,3,4)||||||||||||25200||5040 |- style="text-align:center;" |44||{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}||t_1,2,3,5||[[Bistericantitruncated 7-simplex]] (bacogro)||(0,0,1,1,2,3,4,4)||||||||||||22680||5040 |- style="text-align:center;" |45||{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}||t_0,1,4,5||Pentisteritruncated 7-simplex (tecto)||(0,0,1,2,2,2,3,4)||||||||||||15120||3360 |- style="text-align:center;" |46||{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}||t_0,2,4,5||Pentistericantellated 7-simplex (tecro)||(0,0,1,2,2,3,3,4)||||||||||||25200||5040 |- style="text-align:center; background:#e0f0e0;" |47||{{dark mode invert|{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}}}||t_1,2,4,5||[[Bisteriruncitruncated 7-simplex]] (bicpath)||(0,0,1,2,2,3,4,4)||||||||||||20160||5040 |- style="text-align:center;" |48||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}||t_0,3,4,5||Pentisteriruncinated 7-simplex (tacpo)||(0,0,1,2,3,3,3,4)||||||||||||15120||3360 |- style="text-align:center;" |49||{{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t_0,1,2,6||[[Hexicantitruncated 7-simplex]] (pugro)||(0,1,1,1,1,2,3,4)||||||||||||8400||1680 |- style="text-align:center;" |50||{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t_0,1,3,6||[[Hexiruncitruncated 7-simplex]] (pugato)||(0,1,1,1,2,2,3,4)||||||||||||20160||3360 |- style="text-align:center;" |51||{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}||t_0,2,3,6||[[Hexiruncicantellated 7-simplex]] (pugro)||(0,1,1,1,2,3,3,4)||||||||||||16800||3360 |- style="text-align:center;" |52||{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}||t_0,1,4,6||[[Hexisteritruncated 7-simplex]] (pucto)||(0,1,1,2,2,2,3,4)||||||||||||20160||3360 |- style="text-align:center; background:#e0f0e0;" |53||{{dark mode invert|{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}}}||t_0,2,4,6||[[Hexistericantellated 7-simplex]] (pucroh)||(0,1,1,2,2,3,3,4)||||||||||||30240||5040 |- style="text-align:center; background:#e0f0e0;" |54||{{dark mode invert|{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}}}||t_0,1,5,6||[[Hexipentitruncated 7-simplex]] (putath)||(0,1,2,2,2,2,3,4)||||||||||||8400||1680 |- style="text-align:center;" |55||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t_0,1,2,3,4||[[Steriruncicantitruncated 7-simplex]] (gecco)||(0,0,0,1,2,3,4,5)||||||||||||23520||6720 |- style="text-align:center;" |56||{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t_0,1,2,3,5||Pentiruncicantitruncated 7-simplex (tegapo)||(0,0,1,1,2,3,4,5)||||||||||||45360||10080 |- style="text-align:center;" |57||{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t_0,1,2,4,5||Pentistericantitruncated 7-simplex (tecagro)||(0,0,1,2,2,3,4,5)||||||||||||40320||10080 |- style="text-align:center;" |58||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t_0,1,3,4,5||Pentisteriruncitruncated 7-simplex (tacpeto)||(0,0,1,2,3,3,4,5)||||||||||||40320||10080 |- style="text-align:center;" |59||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t_0,2,3,4,5||Pentisteriruncicantellated 7-simplex (tacpro)||(0,0,1,2,3,4,4,5)||||||||||||40320||10080 |- style="text-align:center; background:#e0f0e0;" |60||{{dark mode invert|{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}}}||t_1,2,3,4,5||[[Bisteriruncicantitruncated 7-simplex]] (gabach)||(0,0,1,2,3,4,5,5)||||||||||||35280||10080 |- style="text-align:center;" |61||{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t_0,1,2,3,6||[[Hexiruncicantitruncated 7-simplex]] (pugopo)||(0,1,1,1,2,3,4,5)||||||||||||30240||6720 |- style="text-align:center;" |62||{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t_0,1,2,4,6||[[Hexistericantitruncated 7-simplex]] (pucagro)||(0,1,1,2,2,3,4,5)||||||||||||50400||10080 |- style="text-align:center;" |63||{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t_0,1,3,4,6||[[Hexisteriruncitruncated 7-simplex]] (pucpato)||(0,1,1,2,3,3,4,5)||||||||||||45360||10080 |- style="text-align:center; background:#e0f0e0;" |64||{{dark mode invert|{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}}}||t_0,2,3,4,6||[[Hexisteriruncicantellated 7-simplex]] (pucproh)||(0,1,1,2,3,4,4,5)||||||||||||45360||10080 |- style="text-align:center;" |65||{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t_0,1,2,5,6||[[Hexipenticantitruncated 7-simplex]] (putagro)||(0,1,2,2,2,3,4,5)||||||||||||30240||6720 |- style="text-align:center; background:#e0f0e0;" |66||{{dark mode invert|{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}}}||t_0,1,3,5,6||[[Hexipentiruncitruncated 7-simplex]] (putpath)||(0,1,2,2,3,3,4,5)||||||||||||50400||10080 |- style="text-align:center;" |67||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t_0,1,2,3,4,5||Pentisteriruncicantitruncated 7-simplex (geto)||(0,0,1,2,3,4,5,6)||||||||||||70560||20160 |- style="text-align:center;" |68||{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t_0,1,2,3,4,6||[[Hexisteriruncicantitruncated 7-simplex]] (pugaco)||(0,1,1,2,3,4,5,6)||||||||||||80640||20160 |- style="text-align:center;" |69||{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t_0,1,2,3,5,6||[[Hexipentiruncicantitruncated 7-simplex]] (putgapo)||(0,1,2,2,3,4,5,6)||||||||||||80640||20160 |- style="text-align:center; background:#e0f0e0;" |70||{{dark mode invert|{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}}}||t_0,1,2,4,5,6||[[Hexipentistericantitruncated 7-simplex]] (putcagroh)||(0,1,2,3,3,4,5,6)||||||||||||80640||20160 |- style="text-align:center; background:#e0f0e0;" |71||{{dark mode invert|{{nowrap|{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}}}}}||t_{0,1,2,3,4,5,6}||[[Omnitruncated 7-simplex]] (guph)||(0,1,2,3,4,5,6,7)||||||||||||141120||40320 |}

== The B₇ family == The B₇ family has symmetry of order 645120 (7 [[factorial]] x 2⁷).

There are 127 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. Bowers names and acronym are given for cross-referencing.

See also a [[list of B7 polytopes]] for symmetric [[Coxeter plane]] graphs of these polytopes.

{| class="wikitable collapsible collapsed" !colspan=12|B₇ uniform polytopes |- !rowspan=2|# !rowspan=2|[[Coxeter-Dynkin diagram]]<BR>t-notation !rowspan=2|Name (BSA) !rowspan=2|Base point !colspan=7|Element counts |- !6||5||4||3||2||1||0 |- style="text-align:center; background:#f0e0e0;" !1 |<!-- [x3o3o3o3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1}}}}<BR>t₀{3,3,3,3,3,4}||[[7-orthoplex]] (zee)|||(0,0,0,0,0,0,1)√2||128||448||672||560||280||84||14 |- style="text-align:center; background:#f0e0e0;" !2 |<!-- [o3x3o3o3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node}}}}<BR>t₁{3,3,3,3,3,4}||[[Rectified 7-orthoplex]] (rez)|||(0,0,0,0,0,1,1)√2||142||1344||3360||3920||2520||840||84 |- style="text-align:center; background:#f0e0e0;" !3 |<!-- [o3o3x3o3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node}}}}<BR>t₂{3,3,3,3,3,4}||[[Birectified 7-orthoplex]] (barz)|||(0,0,0,0,1,1,1)√2||142||1428||6048||10640||8960||3360||280 |- style="text-align:center; background:#e0f0e0;" !4 |<!-- [o3o3o3x3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node}}}}<BR>t₃{4,3,3,3,3,3}||[[Trirectified 7-cube]] (sez)|||(0,0,0,1,1,1,1)√2||142||1428||6328||14560||15680||6720||560 |- style="text-align:center; background:#e0e0f0;" !5 |<!-- [o3o3o3o3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node}}}}<BR>t₂{4,3,3,3,3,3}||[[Birectified 7-cube]] (bersa)|||(0,0,1,1,1,1,1)√2||142||1428||5656||11760||13440||6720||672 |- style="text-align:center; background:#e0e0f0;" !6 |<!-- [o3o3o3o3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node}}}}<BR>t₁{4,3,3,3,3,3}||[[Rectified 7-cube]] (rasa)|||(0,1,1,1,1,1,1)√2||142||980||2968||5040||5152||2688||448 |- style="text-align:center; background:#e0e0f0;" !7 |<!-- [o3o3o3o3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}}}<BR>t₀{4,3,3,3,3,3}||[[7-cube]] (hept)|||(0,0,0,0,0,0,0)√2 + (1,1,1,1,1,1,1)||14||84||280||560||672||448||128 |- style="text-align:center; background:#f0e0e0;" !8 |<!-- [x3x3o3o3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}}}<BR>t_0,1{3,3,3,3,3,4}||[[Truncated 7-orthoplex]] (Taz)|||(0,0,0,0,0,1,2)√2||142||1344||3360||4760||2520||924||168 |- style="text-align:center; background:#f0e0e0;" !9 |<!-- [x3o3x3o3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}}}<BR>t_0,2{3,3,3,3,3,4}||[[Cantellated 7-orthoplex]] (Sarz)|||(0,0,0,0,1,1,2)√2||226||4200||15456||24080||19320||7560||840 |- style="text-align:center; background:#f0e0e0;" !10 |<!-- [o3x3x3o3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}}}<BR>t_1,2{3,3,3,3,3,4}||[[Bitruncated 7-orthoplex]] (Botaz)|||(0,0,0,0,1,2,2)√2|||||| || || ||4200||840 |- style="text-align:center; background:#f0e0e0;" !11 |<!-- [x3o3o3x3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}}}<BR>t_0,3{3,3,3,3,3,4}||[[Runcinated 7-orthoplex]] (Spaz)|||(0,0,0,1,1,1,2)√2|||||| || || ||23520||2240 |- style="text-align:center; background:#f0e0e0;" !12 |<!-- [o3x3o3x3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}}}<BR>t_1,3{3,3,3,3,3,4}||[[Bicantellated 7-orthoplex]] (Sebraz)|||(0,0,0,1,1,2,2)√2|||||| || || ||26880||3360 |- style="text-align:center; background:#f0e0e0;" !13 |<!-- [o3o3x3x3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node|3|node}}}}<BR>t_2,3{3,3,3,3,3,4}||[[Tritruncated 7-orthoplex]] (Totaz)|||(0,0,0,1,2,2,2)√2|||||| || || ||10080||2240 |- style="text-align:center; background:#f0e0e0;" !14 |<!-- [x3o3o3o3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}}}<BR>t_0,4{3,3,3,3,3,4}||[[Stericated 7-orthoplex]] (Scaz)|||(0,0,1,1,1,1,2)√2|||||| || || ||33600||3360 |- style="text-align:center; background:#f0e0e0;" !15 |<!-- [o3x3o3o3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}}}<BR>t_1,4{3,3,3,3,3,4}||[[Biruncinated 7-orthoplex]] (Sibpaz)|||(0,0,1,1,1,2,2)√2|||||| || || ||60480||6720 |- style="text-align:center; background:#e0f0e0;" !16 |<!-- [o3o3x3o3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}}}<BR>t_2,4{4,3,3,3,3,3}||[[Tricantellated 7-cube]] (Strasaz)|||(0,0,1,1,2,2,2)√2|||||| || || ||47040||6720 |- style="text-align:center; background:#e0e0f0;" !17 |<!-- [o3o3o3x3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}}}<BR>t_2,3{4,3,3,3,3,3}||[[Tritruncated 7-cube]] (Tatsa)|||(0,0,1,2,2,2,2)√2|||||| || || ||13440||3360 |- style="text-align:center; background:#f0e0e0;" !18 |<!-- [x3o3o3o3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node_1}}}}<BR>t_0,5{3,3,3,3,3,4}||[[Pentellated 7-orthoplex]] (Staz)|||(0,1,1,1,1,1,2)√2|||||| || || ||20160||2688 |- style="text-align:center; background:#e0f0e0;" !19 |<!-- [o3x3o3o3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node|3|node|3|node_1|3|node}}}}<BR>t_1,5{4,3,3,3,3,3}||[[Bistericated 7-cube]] (Sabcosaz)|||(0,1,1,1,1,2,2)√2|||||| || || ||53760||6720 |- style="text-align:center; background:#e0e0f0;" !20 |<!-- [o3o3x3o3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node|3|node}}}}<BR>t_1,4{4,3,3,3,3,3}||[[Biruncinated 7-cube]] (Sibposa)|||(0,1,1,1,2,2,2)√2|||||| || || ||67200||8960 |- style="text-align:center; background:#e0e0f0;" !21 |<!-- [o3o3o3x3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node|3|node}}}}<BR>t_1,3{4,3,3,3,3,3}||[[Bicantellated 7-cube]] (Sibrosa)|||(0,1,1,2,2,2,2)√2|||||| || || ||40320||6720 |- style="text-align:center; background:#e0e0f0;" !22 |<!-- [o3o3o3o3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node}}}}<BR>t_1,2{4,3,3,3,3,3}||[[Bitruncated 7-cube]] (Betsa)|||(0,1,2,2,2,2,2)√2|||||| || || ||9408||2688 |- style="text-align:center; background:#e0f0e0;" !23 |<!-- [x3o3o3o3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node_1}}}}<BR>t_0,6{4,3,3,3,3,3}||[[Hexicated 7-cube]] (Supposaz)|||(0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1)|||||| || || ||5376||896 |- style="text-align:center; background:#e0e0f0;" !24 |<!-- [o3x3o3o3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1|3|node}}}}<BR>t_0,5{4,3,3,3,3,3}||[[Pentellated 7-cube]] (Stesa)|||(0,0,0,0,0,1,1)√2 + (1,1,1,1,1,1,1)|||||| || || ||20160||2688 |- style="text-align:center; background:#e0e0f0;" !25 |<!-- [o3o3x3o3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node|3|node}}}}<BR>t_0,4{4,3,3,3,3,3}||[[Stericated 7-cube]] (Scosa)|||(0,0,0,0,1,1,1)√2 + (1,1,1,1,1,1,1)|||||| || || ||35840||4480 |- style="text-align:center; background:#e0e0f0;" !26 |<!-- [o3o3o3x3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node|3|node}}}}<BR>t_0,3{4,3,3,3,3,3}||[[Runcinated 7-cube]] (Spesa)|||(0,0,0,1,1,1,1)√2 + (1,1,1,1,1,1,1)|||||| || || ||33600||4480 |- style="text-align:center; background:#e0e0f0;" !27 |<!-- [o3o3o3o3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node|3|node}}}}<BR>t_0,2{4,3,3,3,3,3}||[[Cantellated 7-cube]] (Sersa)|||(0,0,1,1,1,1,1)√2 + (1,1,1,1,1,1,1)|||||| || || ||16128||2688 |- style="text-align:center; background:#e0e0f0;" !28 |<!-- [o3o3o3o3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node|3|node}}}}<BR>t_0,1{4,3,3,3,3,3}||[[Truncated 7-cube]] (Tasa)|||(0,1,1,1,1,1,1)√2 + (1,1,1,1,1,1,1)||142||980||2968||5040||5152||3136||896 |- style="text-align:center; background:#f0e0e0;" !29 |<!-- [x3x3x3o3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2{3,3,3,3,3,4}||[[Cantitruncated 7-orthoplex]] (Garz)|||(0,1,2,3,3,3,3)√2|||||| || || ||8400||1680 |- style="text-align:center; background:#f0e0e0;" !30 |<!-- [x3x3o3x3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,3{3,3,3,3,3,4}||[[Runcitruncated 7-orthoplex]] (Potaz)|||(0,1,2,2,3,3,3)√2|||||| || || ||50400||6720 |- style="text-align:center; background:#f0e0e0;" !31 |<!-- [x3o3x3x3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,3{3,3,3,3,3,4}||[[Runcicantellated 7-orthoplex]] (Parz)|||(0,1,1,2,3,3,3)√2|||||| || || ||33600||6720 |- style="text-align:center; background:#f0e0e0;" !32 |<!-- [o3x3x3x3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}}}<BR>t_1,2,3{3,3,3,3,3,4}||[[Bicantitruncated 7-orthoplex]] (Gebraz)|||(0,0,1,2,3,3,3)√2|||||| || || ||30240||6720 |- style="text-align:center; background:#f0e0e0;" !33 |<!-- [x3x3o3o3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,4{3,3,3,3,3,4}||[[Steritruncated 7-orthoplex]] (Catz)|||(0,0,1,1,1,2,3)√2|||||| || || ||107520||13440 |- style="text-align:center; background:#f0e0e0;" !34 |<!-- [x3o3x3o3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,4{3,3,3,3,3,4}||[[Stericantellated 7-orthoplex]] (Craze)|||(0,0,1,1,2,2,3)√2|||||| || || ||141120||20160 |- style="text-align:center; background:#f0e0e0;" !35 |<!-- [o3x3x3o3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}}}<BR>t_1,2,4{3,3,3,3,3,4}||[[Biruncitruncated 7-orthoplex]] (Baptize)|||(0,0,1,1,2,3,3)√2|||||| || || ||120960||20160 |- style="text-align:center; background:#f0e0e0;" !36 |<!-- [x3o3o3x3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}}}<BR>t_0,3,4{3,3,3,3,3,4}||[[Steriruncinated 7-orthoplex]] (Copaz)|||(0,1,1,1,2,3,3)√2|||||| || || ||67200||13440 |- style="text-align:center; background:#f0e0e0;" !37 |<!-- [o3x3o3x3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}}}<BR>t_1,3,4{3,3,3,3,3,4}||[[Biruncicantellated 7-orthoplex]] (Boparz)|||(0,0,1,2,2,3,3)√2|||||| || || ||100800||20160 |- style="text-align:center; background:#e0f0e0;" !38 |<!-- [o3o3x3x3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}}}<BR>t_2,3,4{4,3,3,3,3,3}||[[Tricantitruncated 7-cube]] (Gotrasaz)|||(0,0,0,1,2,3,3)√2|||||| || || ||53760||13440 |- style="text-align:center; background:#f0e0e0;" !39 |<!-- [x3x3o3o3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,5{3,3,3,3,3,4}||[[Pentitruncated 7-orthoplex]] (Tetaz)|||(0,1,1,1,1,2,3)√2|||||| || || ||87360||13440 |- style="text-align:center; background:#f0e0e0;" !40 |<!-- [x3o3x3o3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,5{3,3,3,3,3,4}||[[Penticantellated 7-orthoplex]] (Teroz)|||(0,1,1,1,2,2,3)√2|||||| || || ||188160||26880 |- style="text-align:center; background:#f0e0e0;" !41 |<!-- [o3x3x3o3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}}}<BR>t_1,2,5{3,3,3,3,3,4}||[[Bisteritruncated 7-orthoplex]] (Boctaz)|||(0,1,1,1,2,3,3)√2|||||| || || ||147840||26880 |- style="text-align:center; background:#f0e0e0;" !42 |<!-- [x3o3o3x3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}}}<BR>t_0,3,5{3,3,3,3,3,4}||[[Pentiruncinated 7-orthoplex]] (Topaz)|||(0,1,1,2,2,2,3)√2|||||| || || ||174720||26880 |- style="text-align:center; background:#e0f0e0;" !43 |<!-- [o3x3o3x3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}}}<BR>t_1,3,5{4,3,3,3,3,3}||[[Bistericantellated 7-cube]] (Bacresaz)|||(0,1,1,2,2,3,3)√2|||||| || || ||241920||40320 |- style="text-align:center; background:#e0e0f0;" !44 |<!-- [o3o3x3x3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}}}<BR>t_1,3,4{4,3,3,3,3,3}||[[Biruncicantellated 7-cube]] (Bopresa)|||(0,1,1,2,3,3,3)√2|||||| || || ||120960||26880 |- style="text-align:center; background:#f0e0e0;" !45 |<!-- [x3o3o3o3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}}}<BR>t_0,4,5{3,3,3,3,3,4}||[[Pentistericated 7-orthoplex]] (Tocaz)|||(0,1,2,2,2,2,3)√2|||||| || || ||67200||13440 |- style="text-align:center; background:#e0e0f0;" !46 |<!-- [o3x3o3o3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}}}<BR>t_1,2,5{4,3,3,3,3,3}||[[Bisteritruncated 7-cube]] (Bactasa)|||(0,1,2,2,2,3,3)√2|||||| || || ||147840||26880 |- style="text-align:center; background:#e0e0f0;" !47 |<!-- [o3o3x3o3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}}}<BR>t_1,2,4{4,3,3,3,3,3}||[[Biruncitruncated 7-cube]] (Biptesa)|||(0,1,2,2,3,3,3)√2|||||| || || ||134400||26880 |- style="text-align:center; background:#e0e0f0;" !48 |<!-- [o3o3o3x3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}}}<BR>t_1,2,3{4,3,3,3,3,3}||[[Bicantitruncated 7-cube]] (Gibrosa)|||(0,1,2,3,3,3,3)√2|||||| || || ||47040||13440 |- style="text-align:center; background:#f0e0e0;" !49 |<!-- [x3x3o3o3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,6{3,3,3,3,3,4}||[[Hexitruncated 7-orthoplex]] (Putaz)|||(0,0,0,0,0,1,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||29568||5376 |- style="text-align:center; background:#f0e0e0;" !50 |<!-- [x3o3x3o3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,6{3,3,3,3,3,4}||[[Hexicantellated 7-orthoplex]] (Puraz)|||(0,0,0,0,1,1,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||94080||13440 |- style="text-align:center; background:#e0e0f0;" !51 |<!-- [o3x3x3o3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}}}<BR>t_0,4,5{4,3,3,3,3,3}||[[Pentistericated 7-cube]] (Tacosa)|||(0,0,0,0,1,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||67200||13440 |- style="text-align:center; background:#e0f0e0;" !52 |<!-- [x3o3o3x3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}}}<BR>t_0,3,6{4,3,3,3,3,3}||[[Hexiruncinated 7-cube]] (Pupsez)|||(0,0,0,1,1,1,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||134400||17920 |- style="text-align:center; background:#e0e0f0;" !53 |<!-- [o3x3o3x3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}}}<BR>t_0,3,5{4,3,3,3,3,3}||[[Pentiruncinated 7-cube]] (Tapsa)|||(0,0,0,1,1,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||174720||26880 |- style="text-align:center; background:#e0e0f0;" !54 |<!-- [o3o3x3x3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node|3|node}}}}<BR>t_0,3,4{4,3,3,3,3,3}||[[Steriruncinated 7-cube]] (Capsa)|||(0,0,0,1,2,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||80640||17920 |- style="text-align:center; background:#e0e0f0;" !55 |<!-- [x3o3o3o3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}}}<BR>t_0,2,6{4,3,3,3,3,3}||[[Hexicantellated 7-cube]] (Purosa)|||(0,0,1,1,1,1,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||94080||13440 |- style="text-align:center; background:#e0e0f0;" !56 |<!-- [o3x3o3o3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}}}<BR>t_0,2,5{4,3,3,3,3,3}||[[Penticantellated 7-cube]] (Tersa)|||(0,0,1,1,1,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||188160||26880 |- style="text-align:center; background:#e0e0f0;" !57 |<!-- [o3o3x3o3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}}}<BR>t_0,2,4{4,3,3,3,3,3}||[[Stericantellated 7-cube]] (Carsa)|||(0,0,1,1,2,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||161280||26880 |- style="text-align:center; background:#e0e0f0;" !58 |<!-- [o3o3o3x3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}}}<BR>t_0,2,3{4,3,3,3,3,3}||[[Runcicantellated 7-cube]] (Parsa)|||(0,0,1,2,2,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||53760||13440 |- style="text-align:center; background:#e0e0f0;" !59 |<!-- [x3o3o3o3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node|3|node_1}}}}<BR>t_0,1,6{4,3,3,3,3,3}||[[Hexitruncated 7-cube]] (Putsa)|||(0,1,1,1,1,1,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||29568||5376 |- style="text-align:center; background:#e0e0f0;" !60 |<!-- [o3x3o3o3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1|3|node}}}}<BR>t_0,1,5{4,3,3,3,3,3}||[[Pentitruncated 7-cube]] (Tetsa)|||(0,1,1,1,1,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||87360||13440 |- style="text-align:center; background:#e0e0f0;" !61 |<!-- [o3o3x3o3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node|3|node}}}}<BR>t_0,1,4{4,3,3,3,3,3}||[[Steritruncated 7-cube]] (Catsa)|||(0,1,1,1,2,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||116480||17920 |- style="text-align:center; background:#e0e0f0;" !62 |<!-- [o3o3o3x3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node|3|node}}}}<BR>t_0,1,3{4,3,3,3,3,3}||[[Runcitruncated 7-cube]] (Petsa)|||(0,1,1,2,2,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||73920||13440 |- style="text-align:center; background:#e0e0f0;" !63 |<!-- [o3o3o3o3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node|3|node}}}}<BR>t_0,1,2{4,3,3,3,3,3}||[[Cantitruncated 7-cube]] (Gersa)|||(0,1,2,2,2,2,2)√2 + (1,1,1,1,1,1,1)|||||| || || ||18816||5376 |- style="text-align:center; background:#f0e0e0;" !64 |<!-- [x3x3x3x3o3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,3{3,3,3,3,3,4}||[[Runcicantitruncated 7-orthoplex]] (Gopaz)|||(0,1,2,3,4,4,4)√2|||||| || || ||60480||13440 |- style="text-align:center; background:#f0e0e0;" !65 |<!-- [x3x3x3o3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,4{3,3,3,3,3,4}||[[Stericantitruncated 7-orthoplex]] (Cogarz)|||(0,0,1,1,2,3,4)√2|||||| || || ||241920||40320 |- style="text-align:center; background:#f0e0e0;" !66 |<!-- [x3x3o3x3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,3,4{3,3,3,3,3,4}||[[Steriruncitruncated 7-orthoplex]] (Captaz)|||(0,0,1,2,2,3,4)√2|||||| || || ||181440||40320 |- style="text-align:center; background:#f0e0e0;" !67 |<!-- [x3o3x3x3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,3,4{3,3,3,3,3,4}||[[Steriruncicantellated 7-orthoplex]] (Caparz)|||(0,0,1,2,3,3,4)√2|||||| || || ||181440||40320 |- style="text-align:center; background:#f0e0e0;" !68 |<!-- [o3x3x3x3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}}}<BR>t_1,2,3,4{3,3,3,3,3,4}||[[Biruncicantitruncated 7-orthoplex]] (Gibpaz)|||(0,0,1,2,3,4,4)√2|||||| || || ||161280||40320 |- style="text-align:center; background:#f0e0e0;" !69 |<!-- [x3x3x3o3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,5{3,3,3,3,3,4}||[[Penticantitruncated 7-orthoplex]] (Tograz)|||(0,1,1,1,2,3,4)√2|||||| || || ||295680||53760 |- style="text-align:center; background:#f0e0e0;" !70 |<!-- [x3x3o3x3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,3,5{3,3,3,3,3,4}||[[Pentiruncitruncated 7-orthoplex]] (Toptaz)|||(0,1,1,2,2,3,4)√2|||||| || || ||443520||80640 |- style="text-align:center; background:#f0e0e0;" !71 |<!-- [x3o3x3x3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,3,5{3,3,3,3,3,4}||[[Pentiruncicantellated 7-orthoplex]] (Toparz)|||(0,1,1,2,3,3,4)√2|||||| || || ||403200||80640 |- style="text-align:center; background:#f0e0e0;" !72 |<!-- [o3x3x3x3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}}}<BR>t_1,2,3,5{3,3,3,3,3,4}||[[Bistericantitruncated 7-orthoplex]] (Becogarz)|||(0,1,1,2,3,4,4)√2|||||| || || ||362880||80640 |- style="text-align:center; background:#f0e0e0;" !73 |<!-- [x3x3o3o3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,4,5{3,3,3,3,3,4}||[[Pentisteritruncated 7-orthoplex]] (Tacotaz)|||(0,1,2,2,2,3,4)√2|||||| || || ||241920||53760 |- style="text-align:center; background:#f0e0e0;" !74 |<!-- [x3o3x3o3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,4,5{3,3,3,3,3,4}||[[Pentistericantellated 7-orthoplex]] (Tocarz)|||(0,1,2,2,3,3,4)√2|||||| || || ||403200||80640 |- style="text-align:center; background:#e0f0e0;" !75 |<!-- [o3x3x3o3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}}}<BR>t_1,2,4,5{4,3,3,3,3,3}||[[Bisteriruncitruncated 7-cube]] (Bocaptosaz)|||(0,1,2,2,3,4,4)√2|||||| |||| ||322560||80640 |- style="text-align:center; background:#f0e0e0;" !76 |<!-- [x3o3o3x3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}}}<BR>t_0,3,4,5{3,3,3,3,3,4}||[[Pentisteriruncinated 7-orthoplex]] (Tecpaz)|||(0,1,2,3,3,3,4)√2|||||| || || ||241920||53760 |- style="text-align:center; background:#e0e0f0;" !77 |<!-- [o3x3o3x3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}}}<BR>t_1,2,3,5{4,3,3,3,3,3}||[[Bistericantitruncated 7-cube]] (Becgresa)|||(0,1,2,3,3,4,4)√2|||||| || || ||362880||80640 |- style="text-align:center; background:#e0e0f0;" !78 |<!-- [o3o3x3x3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}}}<BR>t_1,2,3,4{4,3,3,3,3,3}||[[Biruncicantitruncated 7-cube]] (Gibposa)|||(0,1,2,3,4,4,4)√2|||||| || || ||188160||53760 |- style="text-align:center; background:#f0e0e0;" !79 |<!-- [x3x3x3o3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,6{3,3,3,3,3,4}||[[Hexicantitruncated 7-orthoplex]] (Pugarez)|||(0,0,0,0,1,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||134400||26880 |- style="text-align:center; background:#f0e0e0;" !80 |<!-- [x3x3o3x3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,3,6{3,3,3,3,3,4}||[[Hexiruncitruncated 7-orthoplex]] (Papataz)|||(0,0,0,1,1,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||322560||53760 |- style="text-align:center; background:#f0e0e0;" !81 |<!-- [x3o3x3x3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,3,6{3,3,3,3,3,4}||[[Hexiruncicantellated 7-orthoplex]] (Puparez)|||(0,0,0,1,2,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||268800||53760 |- style="text-align:center; background:#e0e0f0;" !82 |<!-- [o3x3x3x3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}}}<BR>t_0,3,4,5{4,3,3,3,3,3}||[[Pentisteriruncinated 7-cube]] (Tecpasa)|||(0,0,0,1,2,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||241920||53760 |- style="text-align:center; background:#f0e0e0;" !83 |<!-- [x3x3o3o3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,4,6{3,3,3,3,3,4}||[[Hexisteritruncated 7-orthoplex]] (Pucotaz)|||(0,0,1,1,1,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||322560||53760 |- style="text-align:center; background:#e0f0e0;" !84 |<!-- [x3o3x3o3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,4,6{4,3,3,3,3,3}||[[Hexistericantellated 7-cube]] (Pucrosaz)|||(0,0,1,1,2,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||483840||80640 |- style="text-align:center; background:#e0e0f0;" !85 |<!-- [o3x3x3o3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}}}<BR>t_0,2,4,5{4,3,3,3,3,3}||[[Pentistericantellated 7-cube]] (Tecresa)|||(0,0,1,1,2,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||403200||80640 |- style="text-align:center; background:#e0e0f0;" !86 |<!-- [x3o3o3x3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}}}<BR>t_0,2,3,6{4,3,3,3,3,3}||[[Hexiruncicantellated 7-cube]] (Pupresa)|||(0,0,1,2,2,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||268800||53760 |- style="text-align:center; background:#e0e0f0;" !87 |<!-- [o3x3o3x3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}}}<BR>t_0,2,3,5{4,3,3,3,3,3}||[[Pentiruncicantellated 7-cube]] (Topresa)|||(0,0,1,2,2,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||403200||80640 |- style="text-align:center; background:#e0e0f0;" !88 |<!-- [o3o3x3x3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}}}<BR>t_0,2,3,4{4,3,3,3,3,3}||[[Steriruncicantellated 7-cube]] (Copresa)|||(0,0,1,2,3,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||215040||53760 |- style="text-align:center; background:#e0f0e0;" !89 |<!-- [x3x3o3o3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,5,6{4,3,3,3,3,3}||[[Hexipentitruncated 7-cube]] (Putatosez)|||(0,1,1,1,1,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||134400||26880 |- style="text-align:center; background:#e0e0f0;" !90 |<!-- [x3o3x3o3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}}}<BR>t_0,1,4,6{4,3,3,3,3,3}||[[Hexisteritruncated 7-cube]] (Pacutsa)|||(0,1,1,1,2,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||322560||53760 |- style="text-align:center; background:#e0e0f0;" !91 |<!-- [o3x3x3o3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}}}<BR>t_0,1,4,5{4,3,3,3,3,3}||[[Pentisteritruncated 7-cube]] (Tecatsa)|||(0,1,1,1,2,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||241920||53760 |- style="text-align:center; background:#e0e0f0;" !92 |<!-- [x3o3o3x3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}}}<BR>t_0,1,3,6{4,3,3,3,3,3}||[[Hexiruncitruncated 7-cube]] (Pupetsa)|||(0,1,1,2,2,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||322560||53760 |- style="text-align:center; background:#e0e0f0;" !93 |<!-- [o3x3o3x3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}}}<BR>t_0,1,3,5{4,3,3,3,3,3}||[[Pentiruncitruncated 7-cube]] (Toptosa)|||(0,1,1,2,2,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||443520||80640 |- style="text-align:center; background:#e0e0f0;" !94 |<!-- [o3o3x3x3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}}}<BR>t_0,1,3,4{4,3,3,3,3,3}||[[Steriruncitruncated 7-cube]] (Captesa)|||(0,1,1,2,3,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||215040||53760 |- style="text-align:center; background:#e0e0f0;" !95 |<!-- [x3o3o3o3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}}}<BR>t_0,1,2,6{4,3,3,3,3,3}||[[Hexicantitruncated 7-cube]] (Pugrosa)|||(0,1,2,2,2,2,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||134400||26880 |- style="text-align:center; background:#e0e0f0;" !96 |<!-- [o3x3o3o3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}}}<BR>t_0,1,2,5{4,3,3,3,3,3}||[[Penticantitruncated 7-cube]] (Togresa)|||(0,1,2,2,2,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||295680||53760 |- style="text-align:center; background:#e0e0f0;" !97 |<!-- [o3o3x3o3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}}}<BR>t_0,1,2,4{4,3,3,3,3,3}||[[Stericantitruncated 7-cube]] (Cogarsa)|||(0,1,2,2,3,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||268800||53760 |- style="text-align:center; background:#e0e0f0;" !98 |<!-- [o3o3o3x3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}}}<BR>t_0,1,2,3{4,3,3,3,3,3}||[[Runcicantitruncated 7-cube]] (Gapsa)|||(0,1,2,3,3,3,3)√2 + (1,1,1,1,1,1,1)|||||| || || ||94080||26880 |- style="text-align:center; background:#f0e0e0;" !99 |<!-- [x3x3x3x3x3o4o] -->{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,3,4{3,3,3,3,3,4}||[[Steriruncicantitruncated 7-orthoplex]] (Gocaz)|||(0,0,1,2,3,4,5)√2|||||| || || ||322560||80640 |- style="text-align:center; background:#f0e0e0;" !100 |<!-- [x3x3x3x3o3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,3,5{3,3,3,3,3,4}||[[Pentiruncicantitruncated 7-orthoplex]] (Tegopaz)|||(0,1,1,2,3,4,5)√2|||||| || || ||725760||161280 |- style="text-align:center; background:#f0e0e0;" !101 |<!-- [x3x3x3o3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,4,5{3,3,3,3,3,4}||[[Pentistericantitruncated 7-orthoplex]] (Tecagraz)|||(0,1,2,2,3,4,5)√2|||||| || || ||645120||161280 |- style="text-align:center; background:#f0e0e0;" !102 |<!-- [x3x3o3x3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,3,4,5{3,3,3,3,3,4}||[[Pentisteriruncitruncated 7-orthoplex]] (Tecpotaz)|||(0,1,2,3,3,4,5)√2|||||| || || ||645120||161280 |- style="text-align:center; background:#f0e0e0;" !103 |<!-- [x3o3x3x3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,3,4,5{3,3,3,3,3,4}||[[Pentisteriruncicantellated 7-orthoplex]] (Tacparez)|||(0,1,2,3,4,4,5)√2|||||| || || ||645120||161280 |- style="text-align:center; background:#e0f0e0;" !104 |<!-- [o3x3x3x3x3x4o] -->{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}}}<BR>t_1,2,3,4,5{4,3,3,3,3,3}||[[Bisteriruncicantitruncated 7-cube]] (Gabcosaz)|||(0,1,2,3,4,5,5)√2|||||| || || ||564480||161280 |- style="text-align:center; background:#f0e0e0;" !105 |<!-- [x3x3x3x3o3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,3,6{3,3,3,3,3,4}||[[Hexiruncicantitruncated 7-orthoplex]] (Pugopaz)|||(0,0,0,1,2,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||483840||107520 |- style="text-align:center; background:#f0e0e0;" !106 |<!-- [x3x3x3o3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,4,6{3,3,3,3,3,4}||[[Hexistericantitruncated 7-orthoplex]] (Pucagraz)|||(0,0,1,1,2,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||806400||161280 |- style="text-align:center; background:#f0e0e0;" !107 |<!-- [x3x3o3x3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,3,4,6{3,3,3,3,3,4}||[[Hexisteriruncitruncated 7-orthoplex]] (Pucpotaz)|||(0,0,1,2,2,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||725760||161280 |- style="text-align:center; background:#e0f0e0;" !108 |<!-- [x3o3x3x3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}}}<BR>t_0,2,3,4,6{4,3,3,3,3,3}||[[Hexisteriruncicantellated 7-cube]] (Pucprosaz)|||(0,0,1,2,3,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||725760||161280 |- style="text-align:center; background:#e0e0f0;" !109 |<!-- [o3x3x3x3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}}}<BR>t_0,2,3,4,5{4,3,3,3,3,3}||[[Pentisteriruncicantellated 7-cube]] (Tocpresa)|||(0,0,1,2,3,4,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||645120||161280 |- style="text-align:center; background:#f0e0e0;" !110 |<!-- [x3x3x3o3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,5,6{3,3,3,3,3,4}||[[Hexipenticantitruncated 7-orthoplex]] (Putegraz)|||(0,1,1,1,2,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||483840||107520 |- style="text-align:center; background:#e0f0e0;" !111 |<!-- [x3x3o3x3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,3,5,6{4,3,3,3,3,3}||[[Hexipentiruncitruncated 7-cube]] (Putpetsaz)|||(0,1,1,2,2,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||806400||161280 |- style="text-align:center; background:#e0e0f0;" !112 |<!-- [x3o3x3x3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}}}<BR>t_0,1,3,4,6{4,3,3,3,3,3}||[[Hexisteriruncitruncated 7-cube]] (Pucpetsa)|||(0,1,1,2,3,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||725760||161280 |- style="text-align:center; background:#e0e0f0;" !113 |<!-- [o3x3x3x3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}}}<BR>t_0,1,3,4,5{4,3,3,3,3,3}||[[Pentisteriruncitruncated 7-cube]] (Tecpetsa)|||(0,1,1,2,3,4,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||645120||161280 |- style="text-align:center; background:#e0e0f0;" !114 |<!-- [x3x3o3o3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,2,5,6{4,3,3,3,3,3}||[[Hexipenticantitruncated 7-cube]] (Putgresa)|||(0,1,2,2,2,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||483840||107520 |- style="text-align:center; background:#e0e0f0;" !115 |<!-- [x3o3x3o3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}}}<BR>t_0,1,2,4,6{4,3,3,3,3,3}||[[Hexistericantitruncated 7-cube]] (Pucagrosa)|||(0,1,2,2,3,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||806400||161280 |- style="text-align:center; background:#e0e0f0;" !116 |<!-- [o3x3x3o3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}}}<BR>t_0,1,2,4,5{4,3,3,3,3,3}||[[Pentistericantitruncated 7-cube]] (Tecgresa)|||(0,1,2,2,3,4,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||645120||161280 |- style="text-align:center; background:#e0e0f0;" !117 |<!-- [x3o3o3x3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}}}<BR>t_0,1,2,3,6{4,3,3,3,3,3}||[[Hexiruncicantitruncated 7-cube]] (Pugopsa)|||(0,1,2,3,3,3,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||483840||107520 |- style="text-align:center; background:#e0e0f0;" !118 |<!-- [o3x3o3x3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}}}<BR>t_0,1,2,3,5{4,3,3,3,3,3}||[[Pentiruncicantitruncated 7-cube]] (Togapsa)|||(0,1,2,3,3,4,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||725760||161280 |- style="text-align:center; background:#e0e0f0;" !119 |<!-- [o3o3x3x3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}}}<BR>t_0,1,2,3,4{4,3,3,3,3,3}||[[Steriruncicantitruncated 7-cube]] (Gacosa)|||(0,1,2,3,4,4,4)√2 + (1,1,1,1,1,1,1)|||||| || || ||376320||107520 |- style="text-align:center; background:#f0e0e0;" !120 |<!-- [x3x3x3x3x3x4o] -->{{dark mode invert|{{CDD|node|4|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{3,3,3,3,3,4}||[[Pentisteriruncicantitruncated 7-orthoplex]] (Gotaz)|||(0,1,2,3,4,5,6)√2|||||| || || ||1128960||322560 |- style="text-align:center; background:#f0e0e0;" !121 |<!-- [x3x3x3x3x3o4x] -->{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,3,4,6{3,3,3,3,3,4}||[[Hexisteriruncicantitruncated 7-orthoplex]] (Pugacaz)|||(0,0,1,2,3,4,5)√2 + (1,1,1,1,1,1,1)|||||| || || ||1290240||322560 |- style="text-align:center; background:#f0e0e0;" !122 |<!-- [x3x3x3x3o3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,3,5,6{3,3,3,3,3,4}||[[Hexipentiruncicantitruncated 7-orthoplex]] (Putgapaz)|||(0,1,1,2,3,4,5)√2 + (1,1,1,1,1,1,1)|||||| || || ||1290240||322560 |- style="text-align:center; background:#e0f0e0;" !123 |<!-- [x3x3x3o3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}}}<BR>t_0,1,2,4,5,6{4,3,3,3,3,3}||[[Hexipentistericantitruncated 7-cube]] (Putcagrasaz)|||(0,1,2,2,3,4,5)√2 + (1,1,1,1,1,1,1)|||||| || || ||1290240||322560 |- style="text-align:center; background:#e0e0f0;" !124 |<!-- [x3x3o3x3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}}}<BR>t_0,1,2,3,5,6{4,3,3,3,3,3}||[[Hexipentiruncicantitruncated 7-cube]] (Putgapsa)|||(0,1,2,3,3,4,5)√2 + (1,1,1,1,1,1,1)|||||| || || ||1290240||322560 |- style="text-align:center; background:#e0e0f0;" !125 |<!-- [x3o3x3x3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}}}<BR>t_0,1,2,3,4,6{4,3,3,3,3,3}||[[Hexisteriruncicantitruncated 7-cube]] (Pugacasa)|||(0,1,2,3,4,4,5)√2 + (1,1,1,1,1,1,1)|||||| || || ||1290240||322560 |- style="text-align:center; background:#e0e0f0;" !126 |<!-- [o3x3x3x3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}}}<BR>t_0,1,2,3,4,5{4,3,3,3,3,3}||[[Pentisteriruncicantitruncated 7-cube]] (Gotesa)|||(0,1,2,3,4,5,5)√2 + (1,1,1,1,1,1,1)|||||| || || ||1128960||322560 |- style="text-align:center; background:#e0f0e0;" !127 |<!-- [x3x3x3x3x3x4x] -->{{dark mode invert|{{CDD|node_1|4|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}{4,3,3,3,3,3}||[[Omnitruncated 7-cube]] (Guposaz)|||(0,1,2,3,4,5,6)√2 + (1,1,1,1,1,1,1)|| || || || || ||2257920||645120 |}

== The D₇ family == The D₇ family has symmetry of order 322560 (7 [[factorial]] × 2⁶).

This family has 3 × 32 − 1 = 95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D₇ [[Coxeter-Dynkin diagram]]. Of these, 63 (2 × 32 − 1) are repeated from the B₇ family and 32 are unique to this family, listed below. Bowers names and acronym are given for cross-referencing.

See also [[list of D7 polytopes]] for Coxeter plane graphs of these polytopes. {| class="wikitable collapsible collapsed" !colspan=12|D₇ uniform polytopes |- !rowspan=2|# !rowspan=2|[[Coxeter diagram]] !rowspan=2|Names !rowspan=2|Base point<BR>(Alternately signed) !colspan=7|Element counts |- !6||5||4||3||2||1||0 |- align=center !1 ||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node}}||[[7-cube]]<BR>demihepteract (hesa)||(1,1,1,1,1,1,1)||78||532||1624||2800||2240||672||64 |- align=center !2 ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node}}||[[cantic 7-cube]]<BR>truncated demihepteract (thesa)||(1,1,3,3,3,3,3)||142||1428||5656||11760||13440||7392||1344 |- align=center !3 ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node}}||[[runcic 7-cube]]<BR>small rhombated demihepteract (sirhesa)||(1,1,1,3,3,3,3)|| || || || || ||16800||2240 |- align=center !4 ||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node}}||[[steric 7-cube]]<BR>small prismated demihepteract (sphosa)||(1,1,1,1,3,3,3)|| || || || || ||20160||2240 |- align=center !5 ||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node}}||[[pentic 7-cube]]<BR>small cellated demihepteract (sochesa)||(1,1,1,1,1,3,3)|| || || || || ||13440||1344 |- align=center !6 ||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node_1}}||[[hexic 7-cube]]<BR>small terated demihepteract (suthesa)||(1,1,1,1,1,1,3)|| || || || || ||4704||448 |- align=center !7 ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}||[[runcicantic 7-cube]]<BR>great rhombated demihepteract (girhesa)||(1,1,3,5,5,5,5)|| || || || || ||23520||6720 |- align=center !8 ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}||[[stericantic 7-cube]]<BR>prismatotruncated demihepteract (pothesa)||(1,1,3,3,5,5,5)|| || || || || ||73920||13440 |- align=center !9 ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node}}||[[steriruncic 7-cube]]<BR>prismatorhomated demihepteract (prohesa)||(1,1,1,3,5,5,5)|| || || || || ||40320||8960 |- align=center !10 ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}||[[penticantic 7-cube]]<BR>cellitruncated demihepteract (cothesa)||(1,1,3,3,3,5,5)|| || || || || ||87360||13440 |- align=center !11 ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}||[[pentiruncic 7-cube]]<BR>cellirhombated demihepteract (crohesa)||(1,1,1,3,3,5,5)|| || || || || ||87360||13440 |- align=center !12 ||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}||[[pentisteric 7-cube]]<BR>celliprismated demihepteract (caphesa)||(1,1,1,1,3,5,5)|| || || || || ||40320||6720 |- align=center !13 ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}||[[hexicantic 7-cube]]<BR>tericantic demihepteract (tuthesa)||(1,1,3,3,3,3,5)|| || || || || ||43680||6720 |- align=center !14 ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}||[[hexiruncic 7-cube]]<BR>terirhombated demihepteract (turhesa)||(1,1,1,3,3,3,5)|| || || || || ||67200||8960 |- align=center !15 ||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}||[[hexisteric 7-cube]]<BR>teriprismated demihepteract (tuphesa)||(1,1,1,1,3,3,5)|| || || || || ||53760||6720 |- align=center !16 ||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}||[[hexipentic 7-cube]]<BR>tericellated demihepteract (tuchesa)||(1,1,1,1,1,3,5)|| || || || || ||21504||2688 |- align=center !17 ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}||[[steriruncicantic 7-cube]]<BR>great prismated demihepteract (gephosa)||(1,1,3,5,7,7,7)|| || || || || ||94080||26880 |- align=center !18 ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}||[[pentiruncicantic 7-cube]]<BR>celligreatorhombated demihepteract (cagrohesa)||(1,1,3,5,5,7,7)|| || || || || ||181440||40320 |- align=center !19 ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}||[[pentistericantic 7-cube]]<BR>celliprismatotruncated demihepteract (capthesa)||(1,1,3,3,5,7,7)|| || || || || ||181440||40320 |- align=center !20 ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}||[[pentisteriruncic 7-cube]]<BR>celliprismatorhombated demihepteract (coprahesa)||(1,1,1,3,5,7,7)|| || || || || ||120960||26880 |- align=center !21 ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}||[[hexiruncicantic 7-cube]]<BR>terigreatorhombated demihepteract (tugrohesa)||(1,1,3,5,5,5,7)|| || || || || ||120960||26880 |- align=center !22 ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}||[[hexistericantic 7-cube]]<BR>teriprismatotruncated demihepteract (tupthesa)||(1,1,3,3,5,5,7)|| || || || || ||221760||40320 |- align=center !23 ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}||[[hexisteriruncic 7-cube]]<BR>teriprismatorhombated demihepteract (tuprohesa)||(1,1,1,3,5,5,7)|| || || || || ||134400||26880 |- align=center !24 ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}||[[hexipenticantic 7-cube]]<BR>tericellitruncated demihepteract (tucothesa)||(1,1,3,3,3,5,7)|| || || || || ||147840||26880 |- align=center !25 ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}||[[hexipentiruncic 7-cube]]<BR>tericellirhombated demihepteract (tucrohesa)||(1,1,1,3,3,5,7)|| || || || || ||161280||26880 |- align=center !26 ||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}||[[hexipentisteric 7-cube]]<BR>tericelliprismated demihepteract (tucophesa)||(1,1,1,1,3,5,7)|| || || || || ||80640||13440 |- align=center !27 ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}||[[pentisteriruncicantic 7-cube]]<BR>great cellated demihepteract (gochesa)||(1,1,3,5,7,9,9)|| || || || || ||282240||80640 |- align=center !28 ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||[[hexisteriruncicantic 7-cube]]<BR>terigreatoprismated demihepteract (tugphesa)||(1,1,3,5,7,7,9)|| || || || || ||322560||80640 |- align=center !29 ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||[[hexipentiruncicantic 7-cube]]<BR>tericelligreatorhombated demihepteract (tucagrohesa)||(1,1,3,5,5,7,9)|| || || || || ||322560||80640 |- align=center !30 ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||[[hexipentistericantic 7-cube]]<BR>tericelliprismatotruncated demihepteract (tucpathesa)||(1,1,3,3,5,7,9)|| || || || || ||362880||80640 |- align=center !31 ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||[[hexipentisteriruncic 7-cube]]<BR>tericelliprismatorhombated demihepteract (tucprohesa)||(1,1,1,3,5,7,9)|| || || || || ||241920||53760 |- align=center !32 ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} = {{nowrap|{{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}}}||[[hexipentisteriruncicantic 7-cube]]<BR>great terated demihepteract (guthesa)||(1,1,3,5,7,9,11)|| || || || || ||564480||161280 |}

== The E₇ family == The E₇ [[Coxeter group]] has order 2,903,040.

There are 127 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. Bowers names and acronym are given for cross-referencing.

See also a [[list of E7 polytopes]] for symmetric Coxeter plane graphs of these polytopes.

{| class="wikitable collapsible collapsed" !colspan=12|E₇ uniform polytopes |- !rowspan=2|# !rowspan=2|[[Coxeter-Dynkin diagram]]<br /> !rowspan=2|Names !colspan=7|Element counts |- ! 6|| 5|| 4|| 3|| 2|| 1|| 0

|- style="text-align:center;" |1||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||[[Gosset 2 31 polytope|2₃₁]] (laq)||632||4788||16128||20160||10080||2016||126 |- style="text-align:center;" |2||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||[[Rectified 2 31 polytope|Rectified 2₃₁]] (rolaq)||758||10332||47880||100800||90720||30240||2016 |- style="text-align:center;" |3||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||[[Rectified 1 32 polytope|Rectified 1₃₂]] (rolin)||758||12348||72072||191520||241920||120960||10080 |- style="text-align:center;" |4||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||[[1 32 polytope|1₃₂]] (lin)||182||4284||23688||50400||40320||10080||576 |- style="text-align:center;" |5||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||[[Birectified 3 21 polytope|Birectified 3₂₁]] (branq)||758||12348||68040||161280||161280||60480||4032 |- style="text-align:center;" |6||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Rectified 3 21 polytope|Rectified 3₂₁]] (ranq)||758||44352||70560||48384||11592||12096||756 |- style="text-align:center;" |7||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Gosset 3 21 polytope|3₂₁]] (naq)||702||6048||12096||10080||4032||756||56 |- align=center |8||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||[[Truncated 2 31 polytope|Truncated 2₃₁]] (talq)||758||10332||47880||100800||90720||32256||4032 |- align=center |9||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||[[Cantellated 2 31 polytope|Cantellated 2₃₁]] (sirlaq)|| || || || || ||131040||20160 |- align=center |10||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||Bitruncated 2₃₁ (botlaq)||||||||||||||30240 |- align=center |11||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small demified 2₃₁ (shilq)||2774||22428||78120||151200||131040||42336||4032 |- align=center |12||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||demirectified 2₃₁ (hirlaq)||||||||||||||12096 |- align=center |13||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||truncated 1₃₂ (tolin)||||||||||||||20160 |- align=center |14||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||small demiprismated 2₃₁ (shiplaq)||||||||||||||20160 |- align=center |15||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||birectified 1₃₂ (berlin)||758||22428||142632||403200||544320||302400||40320 |- align=center |16||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||tritruncated 3₂₁ (totanq)||||||||||||||40320 |- align=center |17||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||demibirectified 3₂₁ (hobranq)||||||||||||||20160 |- align=center |18||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||small cellated 2₃₁ (scalq)||||||||||||||7560 |- align=center |19||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||small biprismated 2₃₁ (sobpalq)||||||||||||||30240 |- align=center |20||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||small birhombated 3₂₁ (sabranq)||||||||||||||60480 |- align=center |21||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||demirectified 3₂₁ (harnaq)||||||||||||||12096 |- align=center |22||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||bitruncated 3₂₁ (botnaq)||||||||||||||12096 |- align=center |23||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||small terated 3₂₁ (stanq)||||||||||||||1512 |- align=center |24||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||small demicellated 3₂₁ (shocanq)||||||||||||||12096 |- align=center |25||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||small prismated 3₂₁ (spanq)||||||||||||||40320 |- align=center |26||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||small demified 3₂₁ (shanq)||||||||||||||4032 |- align=center |27||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||small rhombated 3₂₁ (sranq)||||||||||||||12096 |- align=center |28||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Truncated 3 21 polytope|Truncated 3₂₁]] (tanq)||758||11592||48384||70560||44352||12852||1512 |- align=center |29||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||great rhombated 2₃₁ (girlaq)||||||||||||||60480 |- align=center |30||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||demitruncated 2₃₁ (hotlaq)||||||||||||||24192 |- align=center |31||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||small demirhombated 2₃₁ (sherlaq)||||||||||||||60480 |- align=center |32||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||demibitruncated 2₃₁ (hobtalq)||||||||||||||60480 |- align=center |33||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||demiprismated 2₃₁ (hiptalq)||||||||||||||80640 |- align=center |34||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||demiprismatorhombated 2₃₁ (hiprolaq)||||||||||||||120960 |- align=center |35||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||bitruncated 1₃₂ (batlin)||||||||||||||120960 |- align=center |36||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small prismated 2₃₁ (spalq)||||||||||||||80640 |- align=center |37||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||small rhombated 1₃₂ (sirlin)||||||||||||||120960 |- align=center |38||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||tritruncated 2₃₁ (tatilq)||||||||||||||80640 |- align=center |39||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||cellitruncated 2₃₁ (catalaq)||||||||||||||60480 |- align=center |40||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||cellirhombated 2₃₁ (crilq)||||||||||||||362880 |- align=center |41||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||biprismatotruncated 2₃₁ (biptalq)||||||||||||||181440 |- align=center |42||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small prismated 1₃₂ (seplin)||||||||||||||60480 |- align=center |43||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||small biprismated 3₂₁ (sabipnaq)||||||||||||||120960 |- align=center |44||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||small demibirhombated 3₂₁ (shobranq)||||||||||||||120960 |- align=center |45||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||cellidemiprismated 2₃₁ (chaplaq)||||||||||||||60480 |- align=center |46||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||demibiprismatotruncated 3₂₁ (hobpotanq)||||||||||||||120960 |- align=center |47||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||great birhombated 3₂₁ (gobranq)||||||||||||||120960 |- align=center |48||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||demibitruncated 3₂₁ (hobtanq)||||||||||||||60480 |- align=center |49||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||teritruncated 2₃₁ (totalq)||||||||||||||24192 |- align=center |50||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||terirhombated 2₃₁ (trilq)||||||||||||||120960 |- align=center |51||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||demicelliprismated 3₂₁ (hicpanq)||||||||||||||120960 |- align=center |52||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small teridemified 2₃₁ (sethalq)||||||||||||||24192 |- align=center |53||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||small cellated 3₂₁ (scanq)||||||||||||||60480 |- align=center |54||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||demiprismated 3₂₁ (hipnaq)||||||||||||||80640 |- align=center |55||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||terirhombated 3₂₁ (tranq)||||||||||||||60480 |- align=center |56||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||demicellirhombated 3₂₁ (hocranq)||||||||||||||120960 |- align=center |57||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||prismatorhombated 3₂₁ (pranq)||||||||||||||120960 |- align=center |58||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||small demirhombated 3₂₁ (sharnaq)||||||||||||||60480 |- align=center |59||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||teritruncated 3₂₁ (tetanq)||||||||||||||15120 |- align=center |60||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||demicellitruncated 3₂₁ (hictanq)||||||||||||||60480 |- align=center |61||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||prismatotruncated 3₂₁ (potanq)||||||||||||||120960 |- align=center |62||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||demitruncated 3₂₁ (hotnaq)||||||||||||||24192 |- align=center |63||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||great rhombated 3₂₁ (granq)||||||||||||||24192 |- align=center |64||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great demified 2₃₁ (gahlaq)||||||||||||||120960 |- align=center |65||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||great demiprismated 2₃₁ (gahplaq)||||||||||||||241920 |- align=center |66||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||prismatotruncated 2₃₁ (potlaq)||||||||||||||241920 |- align=center |67||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||prismatorhombated 2₃₁ (prolaq)||||||||||||||241920 |- align=center |68||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||great rhombated 1₃₂ (girlin)||||||||||||||241920 |- align=center |69||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||celligreatorhombated 2₃₁ (cagrilq)||||||||||||||362880 |- align=center |70||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||cellidemitruncated 2₃₁ (chotalq)||||||||||||||241920 |- align=center |71||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||prismatotruncated 1₃₂ (patlin)||||||||||||||362880 |- align=center |72||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||biprismatorhombated 3₂₁ (bipirnaq)||||||||||||||362880 |- align=center |73||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||tritruncated 1₃₂ (tatlin)||||||||||||||241920 |- align=center |74||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||cellidemiprismatorhombated 2₃₁ (chopralq)||||||||||||||362880 |- align=center |75||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||great demibiprismated 3₂₁ (ghobipnaq)||||||||||||||362880 |- align=center |76||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||celliprismated 2₃₁ (caplaq)||||||||||||||241920 |- align=center |77||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||biprismatotruncated 3₂₁ (boptanq)||||||||||||||362880 |- align=center |78||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||great trirhombated 2₃₁ (gatralaq)||||||||||||||241920 |- align=center |79||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||terigreatorhombated 2₃₁ (togrilq)||||||||||||||241920 |- align=center |80||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||teridemitruncated 2₃₁ (thotalq)||||||||||||||120960 |- align=center |81||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||teridemirhombated 2₃₁ (thorlaq)||||||||||||||241920 |- align=center |82||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||celliprismated 3₂₁ (capnaq)||||||||||||||241920 |- align=center |83||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||teridemiprismatotruncated 2₃₁ (thoptalq)||||||||||||||241920 |- align=center |84||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||teriprismatorhombated 3₂₁ (tapronaq)||||||||||||||362880 |- align=center |85||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||demicelliprismatorhombated 3₂₁ (hacpranq)||||||||||||||362880 |- align=center |86||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||teriprismated 2₃₁ (toplaq)||||||||||||||241920 |- align=center |87||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||cellirhombated 3₂₁ (cranq)||||||||||||||362880 |- align=center |88||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||demiprismatorhombated 3₂₁ (hapranq)||||||||||||||241920 |- align=center |89||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||tericellitruncated 2₃₁ (tectalq)||||||||||||||120960 |- align=center |90||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||teriprismatotruncated 3₂₁ (toptanq)||||||||||||||362880 |- align=center |91||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||demicelliprismatotruncated 3₂₁ (hecpotanq)||||||||||||||362880 |- align=center |92||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||teridemitruncated 3₂₁ (thotanq)||||||||||||||120960 |- align=center |93||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||cellitruncated 3₂₁ (catnaq)||||||||||||||241920 |- align=center |94||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||demiprismatotruncated 3₂₁ (hiptanq)||||||||||||||241920 |- align=center |95||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||terigreatorhombated 3₂₁ (tagranq)||||||||||||||120960 |- align=center |96||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||demicelligreatorhombated 3₂₁ (hicgarnq)||||||||||||||241920 |- align=center |97||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||great prismated 3₂₁ (gopanq)||||||||||||||241920 |- align=center |98||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||great demirhombated 3₂₁ (gahranq)||||||||||||||120960 |- align=center |99||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great prismated 2₃₁ (gopalq)||||||||||||||483840 |- align=center |100||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great cellidemified 2₃₁ (gechalq)||||||||||||||725760 |- align=center |101||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||great birhombated 1₃₂ (gebrolin)||||||||||||||725760 |- align=center |102||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||prismatorhombated 1₃₂ (prolin)||||||||||||||725760 |- align=center |103||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||celliprismatorhombated 2₃₁ (caprolaq)||||||||||||||725760 |- align=center |104||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||great biprismated 2₃₁ (gobpalq)||||||||||||||725760 |- align=center |105||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||tericelliprismated 3₂₁ (ticpanq)||||||||||||||483840 |- align=center |106||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||teridemigreatoprismated 2₃₁ (thegpalq)||||||||||||||725760 |- align=center |107||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||teriprismatotruncated 2₃₁ (teptalq)||||||||||||||725760 |- align=center |108||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||teriprismatorhombated 2₃₁ (topralq)||||||||||||||725760 |- align=center |109||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||cellipriemsatorhombated 3₂₁ (copranq)||||||||||||||725760 |- align=center |110||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||tericelligreatorhombated 2₃₁ (tecgrolaq)||||||||||||||725760 |- align=center |111||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||tericellitruncated 3₂₁ (tectanq)||||||||||||||483840 |- align=center |112||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||teridemiprismatotruncated 3₂₁ (thoptanq)||||||||||||||725760 |- align=center |113||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||celliprismatotruncated 3₂₁ (coptanq)||||||||||||||725760 |- align=center |114||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||teridemicelligreatorhombated 3₂₁ (thocgranq)||||||||||||||483840 |- align=center |115||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||terigreatoprismated 3₂₁ (tagpanq)||||||||||||||725760 |- align=center |116||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||great demicellated 3₂₁ (gahcnaq)||||||||||||||725760 |- align=center |117||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||tericelliprismated laq (tecpalq)||||||||||||||483840 |- align=center |118||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||celligreatorhombated 3₂₁ (cogranq)||||||||||||||725760 |- align=center |119||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||great demified 3₂₁ (gahnq)||||||||||||||483840 |- align=center |120||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great cellated 2₃₁ (gocalq)||||||||||||||1451520 |- align=center |121||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||terigreatoprismated 2₃₁ (tegpalq)||||||||||||||1451520 |- align=center |122||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||tericelliprismatotruncated 3₂₁ (tecpotniq)||||||||||||||1451520 |- align=center |123||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||tericellidemigreatoprismated 2₃₁ (techogaplaq)||||||||||||||1451520 |- align=center |124||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||tericelligreatorhombated 3₂₁ (tacgarnq)||||||||||||||1451520 |- align=center |125||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||tericelliprismatorhombated 2₃₁ (tecprolaq)||||||||||||||1451520 |- align=center |126||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||great cellated 3₂₁ (gocanq)||||||||||||||1451520 |- align=center |127||{{nowrap|{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}}}||great terated 3₂₁ (gotanq)||||||||||||||2903040 |}

== Regular and uniform honeycombs == [[File:Coxeter diagram affine rank7 correspondence.png|518px|thumb|Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.]] There are five fundamental affine [[Coxeter groups]] and sixteen prismatic groups that generate regular and uniform tessellations in 6-space: {| class="wikitable" |- !# !colspan=2|[[Coxeter group]] ![[Coxeter diagram]] !Forms |- align=center |1||<math>{\tilde{A}}_6</math>||[3^[7]]||{{CDD|branch|3ab|nodes|3ab|nodes|split2|node}}||17 |- align=center |2||<math>{\tilde{C}}_6</math>||[4,3⁴,4]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|4|node}}||71 |- align=center |3||<math>{\tilde{B}}_6</math>||h[4,3⁴,4]<br />[4,3³,3^1,1]||{{CDD|nodes|split2|node|3|node|3|node|3|node|4|node}}||95 (32 new) |- align=center |4||<math>{\tilde{D}}_6</math>||q[4,3⁴,4]<br />[3^1,1,3²,3^1,1]||{{CDD|nodes|split2|node|3|node|3|node|split1|nodes}}|| 41 (6 new) |- align=center |5||<math>{\tilde{E}}_6</math>||[3^2,2,2]||{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node}}||39 |}

Regular and uniform tessellations include: * <math>{\tilde{A}}_6</math>, 17 forms ** Uniform [[6-simplex honeycomb]]: {3^[7]} {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}} ** Uniform [[Cyclotruncated 6-simplex honeycomb]]: t_0,1{3^[7]} {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch_11}} ** Uniform [[Omnitruncated 6-simplex honeycomb]]: t_{0,1,2,3,4,5,6,7}{3^[7]} {{CDD|node_1|split1|nodes_11|3ab|nodes_11|3ab|nodes_11|3ab|branch_11}} * <math>{\tilde{C}}_6</math>, [4,3⁴,4], 71 forms ** Regular [[6-cube honeycomb]], represented by symbols {4,3⁴,4}, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|4|node}} * <math>{\tilde{B}}_6</math>, [3^1,1,3³,4], 95 forms, 64 shared with <math>{\tilde{C}}_6</math>, 32 new ** Uniform [[6-demicube honeycomb]], represented by symbols h{4,3⁴,4} = {3^1,1,3³,4}, {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|4|node}} = {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|4|node}} * <math>{\tilde{D}}_6</math>, [3^1,1,3²,3^1,1], 41 unique ringed permutations, most shared with <math>{\tilde{B}}_6</math> and <math>{\tilde{C}}_6</math>, and 6 are new. Coxeter calls the first one a [[quarter 6-cubic honeycomb]]. ** {{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|4|node_h1}} ** {{CDD|nodes_10ru|split2|node_1|3|node|3|node|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|4|node_h1}} ** {{CDD|nodes_10ru|split2|node|3|node_1|3|node|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|4|node_h1}} ** {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|4|node_h1}} ** {{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|4|node_h1}} ** {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|4|node_h1}} * <math>{\tilde{E}}_6</math>: [3^2,2,2], 39 forms ** Uniform [[Gosset 2 22 honeycomb|2₂₂ honeycomb]]: represented by symbols {3,3,3^2,2}, {{CDD|node_1|3|node||3|node|split1|nodes|3ab|nodes}} ** Uniform t₄(2₂₂) honeycomb: 4r{3,3,3^2,2}, {{CDD|node|3|node||3|node|split1|nodes|3ab|nodes_11}} ** Uniform 0₂₂₂ honeycomb: {3^2,2,2}, {{CDD|nodes|3ab|nodes|split2|node_1|3|node|3|node}} ** Uniform t₂(0₂₂₂) honeycomb: 2r{3^2,2,2}, {{CDD|nodes_11|3ab|nodes|split2|node|3|node|3|node_1}}

{| class=wikitable |+ Prismatic groups |- !# !colspan=2|[[Coxeter group]] ![[Coxeter-Dynkin diagram]] |- |1||<math>{\tilde{A}}_5</math>x<math>{\tilde{I}}_1</math>||[3^[6],2,∞]||{{CDD|node|split1|nodes|3ab|nodes|split2|node|2|node|infin|node}} |- |2||<math>{\tilde{B}}_5</math>x<math>{\tilde{I}}_1</math>||[4,3,3^1,1,2,∞]||{{CDD|node|4|node|3|node|3|node|3|node|4|node|2|node|infin|node}} |- |3||<math>{\tilde{C}}_5</math>x<math>{\tilde{I}}_1</math>||[4,3³,4,2,∞]||{{CDD|nodes|split2|node|3|node|3|node|4|node|2|node|infin|node}} |- |4||<math>{\tilde{D}}_5</math>x<math>{\tilde{I}}_1</math>||[3^1,1,3,3^1,1,2,∞]||{{CDD|nodes|split2|node|3|node|split1|nodes|2|node|infin|node}} |- |5||<math>{\tilde{A}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3^[5],2,∞,2,∞,2,∞]||{{CDD|branch|3ab|nodes|split2|node|2|node|infin|node|2|node|infin|node}} |- |6||<math>{\tilde{B}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,3^1,1,2,∞,2,∞]||{{CDD|nodes|split2|node|3|node|4|node|2|node|infin|node|2|node|infin|node}} |- |7||<math>{\tilde{C}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,3,4,2,∞,2,∞]||{{CDD|node|4|node|3|node|3|node|4|node|2|node|infin|node|2|node|infin|node}} |- |8||<math>{\tilde{D}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3^1,1,1,1,2,∞,2,∞]||{{CDD|nodes|split2|node|split1|nodes|2|node|infin|node|2|node|infin|node}} |- |9||<math>{\tilde{F}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3,4,3,3,2,∞,2,∞]||{{CDD|node|3|node|4|node|3|node|3|node|2|node|infin|node|2|node|infin|node}} |- |10||<math>{\tilde{C}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,4,2,∞,2,∞,2,∞]||{{CDD|node|4|node|3|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}} |- |11||<math>{\tilde{B}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3^1,1,2,∞,2,∞,2,∞]||{{CDD|nodes|split2|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}} |- |12||<math>{\tilde{A}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3^[4],2,∞,2,∞,2,∞]||{{CDD|branch|3ab|branch|2|node|infin|node|2|node|infin|node|2|node|infin|node}} |- |13||<math>{\tilde{C}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,4,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|4|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}} |- |14||<math>{\tilde{H}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[6,3,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|6|node|3|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}} |- |15||<math>{\tilde{A}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3^[3],2,∞,2,∞,2,∞,2,∞]||{{CDD|node|split1|branch|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}} |- |16||<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[∞,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}} |}

=== Regular and uniform hyperbolic honeycombs === There are no compact hyperbolic Coxeter groups of rank 7, 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 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams. {| class=wikitable |align=right|<math>{\bar{P}}_6</math> = [3,3^[6]]:<BR>{{CDD|node|split1|nodes|3ab|nodes|split2|node|3|node}} |align=right|<math>{\bar{Q}}_6</math> = [3^1,1,3,3^2,1]:<BR>{{CDD|nodea|3a|branch|3a|branch|3a|nodea|3a|nodea}}

|align=right|<math>{\bar{S}}_6</math> = [4,3,3,3^2,1]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|4a|nodea}} |}

== Notes on the Wythoff construction for the uniform 7-polytopes == The reflective 7-dimensional [[uniform polytope]]s are constructed through a [[Wythoff construction]] process, and represented by a [[Coxeter-Dynkin diagram]], where each node represents a mirror. An active mirror is represented by a ringed node. Each combination of active mirrors generates a unique uniform polytope. Uniform polytopes are named in relation to the [[regular polytope]]s in each family. Some families have two regular constructors and thus may be named in two equally valid ways.

Here are the primary operators available for constructing and naming the uniform 7-polytopes.

The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.

{|class="wikitable" |- !Operation !Extended<br/>[[Schläfli symbol]] !width=110|[[Coxeter-Dynkin diagram|Coxeter-<br/>Dynkin<br/>diagram]] !Description |- ! Parent |width=70| t₀{p,q,r,s,t,u} |{{CDD|node_1|p|node|q|node|r|node|s|node|t|node|u|node}} | Any regular 7-polytope |- ! [[Rectification (geometry)|Rectified]] | t₁{p,q,r,s,t,u} |{{CDD|node|p|node_1|q|node|r|node|s|node|t|node|u|node}} |The edges are fully truncated into single points. The 7-polytope now has the combined faces of the parent and dual. |- ! Birectified | t₂{p,q,r,s,t,u} |{{CDD|node|p|node|q|node_1|r|node|s|node|t|node|u|node}} |Birectification reduces [[Cell (geometry)|cells]] to their [[Dual polytope|duals]]. |- ![[Truncation (geometry)|Truncated]] | t_0,1{p,q,r,s,t,u} |{{CDD|node_1|p|node_1|q|node|r|node|s|node|t|node|u|node}} |Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 7-polytope. The 7-polytope has its original faces doubled in sides, and contains the faces of the dual.<br/>[[File:Cube truncation sequence.svg|360px|class=skin-invert]] |- ![[Bitruncated]] | t_1,2{p,q,r,s,t,u} |{{CDD|node|p|node_1|q|node_1|r|node|s|node|t|node|u|node}} |Bitrunction transforms cells to their dual truncation. |- !Tritruncated | t_2,3{p,q,r,s,t,u} |{{CDD|node|p|node|q|node_1|r|node_1|s|node|t|node|u|node}} |Tritruncation transforms 4-faces to their dual truncation. |- ! [[Cantellation (geometry)|Cantellated]] | t_0,2{p,q,r,s,t,u} |{{CDD|node_1|p|node|q|node_1|r|node|s|node|t|node|u|node}} |In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms.<br/>[[File:Cube cantellation sequence.svg|400px]] |- ! Bicantellated | t_1,3{p,q,r,s,t,u} |{{CDD|node|p|node_1|q|node|r|node_1|s|node|t|node|u|node}} |In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms. |- ! [[Runcination (geometry)|Runcinated]] | t_0,3{p,q,r,s,t,u} |{{CDD|node_1|p|node|q|node|r|node_1|s|node|t|node|u|node}} |Runcination reduces cells and creates new cells at the vertices and edges. |- ! Biruncinated | t_1,4{p,q,r,s,t,u} |{{CDD|node|p|node_1|q|node|r|node|s|node_1|t|node|u|node}} |Runcination reduces cells and creates new cells at the vertices and edges. |- ! [[Sterication|Stericated]] | t_0,4{p,q,r,s,t,u} |{{CDD|node_1|p|node|q|node|r|node|s|node_1|t|node|u|node}} |Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps. |- ! Pentellated | t_0,5{p,q,r,s,t,u} |{{CDD|node_1|p|node|q|node|r|node|s|node|t|node_1|u|node}} |Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps. |- ! Hexicated | t_0,6{p,q,r,s,t,u} |{{CDD|node_1|p|node|q|node|r|node|s|node|t|node|u|node_1}} |Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. ([[Expansion (geometry)|expansion]] operation for 7-polytopes) |- ![[Omnitruncation (geometry)|Omnitruncated]] | t_{0,1,2,3,4,5,6}{p,q,r,s,t,u} |{{nowrap|{{CDD|node_1|p|node_1|q|node_1|r|node_1|s|node_1|t|node_1|u|node_1}}}} |All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied. |}

== 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, 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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{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] * [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 * {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa) with acronyms}}

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

[[Category:7-polytopes]]