Uniform 7-polytope

Graphs of three regular and related uniform polytopes

File:7-simplex t0.svg 7-simplex				File:7-simplex t1.svg Rectified 7-simplex				File:7-simplex t01.svg Truncated 7-simplex
File:7-simplex t02.svg Cantellated 7-simplex				File:7-simplex t03.svg Runcinated 7-simplex				File:7-simplex t04.svg Stericated 7-simplex
File:7-simplex t05.svg Pentellated 7-simplex						File:7-simplex t06.svg Hexicated 7-simplex
File:7-cube t6.svg 7-orthoplex				File:7-cube t56.svg Truncated 7-orthoplex				File:7-cube t5.svg Rectified 7-orthoplex
File:7-cube t46.svg Cantellated 7-orthoplex				File:7-cube t36.svg Runcinated 7-orthoplex				File:7-cube t26.svg Stericated 7-orthoplex
File:7-cube t16.svg Pentellated 7-orthoplex				File:7-cube t06.svg Hexicated 7-cube				File:7-cube t05.svg Pentellated 7-cube
File:7-cube t04.svg Stericated 7-cube				File:7-cube t02.svg Cantellated 7-cube				File:7-cube t03.svg Runcinated 7-cube
File:7-cube t0.svg 7-cube				File:7-cube t01.svg Truncated 7-cube				File:7-cube t1.svg Rectified 7-cube
File:7-demicube t0 D7.svg 7-demicube				File:7-demicube t01 D7.svg Cantic 7-cube				File:7-demicube t02 D7.svg Runcic 7-cube
File:7-demicube t03 D7.svg Steric 7-cube				File:7-demicube t04 D7.svg Pentic 7-cube				File:7-demicube t05 D7.svg Hexic 7-cube
File:E7 graph.svg 321				File:Gosset 2 31 polytope.svg 231				File:Gosset 1 32 petrie.svg 132

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6-polytopes.

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 facets around each 4-face. There are exactly three such convex regular 7-polytopes:

1. {3,3,3,3,3,3} - 7-simplex
2. {4,3,3,3,3,3} - 7-cube
3. {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 numbers and torsion coefficients. 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. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.

Uniform 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 diagrams:

#	Coxeter group			Regular and semiregular forms	Uniform count
1	A7	[36]	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	7-simplex - {36}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	71
2	B7	[4,35]	File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	7-cube - {4,35}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 7-orthoplex - {35,4}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png 7-demicube - h{4,35}, File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	127 + 32
3	D7	[33,1,1]	File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	7-demicube, {3,34,1}, File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 7-orthoplex, {34,31,1}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png	95 (0 unique)
4	E7	[33,2,1]	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	321 - File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea 1.png 132 - File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png 231 - File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	127

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 diagrams with one or more rings. All 71 are enumerated below. 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.

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 diagrams with one or more rings. Johnson and Bowers names. See also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes.

The D₇ family

The D₇ family has symmetry of order 322560 (7 factorial x 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.

The E₇ family

The E₇ Coxeter group has order 2,903,040. There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. See also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes.

Regular and uniform honeycombs

File:Coxeter diagram affine rank7 correspondence.png

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:

#	Coxeter group		Coxeter diagram	Forms
1	${\tilde{A}}_6$	[3[7]]	File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png	17
2	${\tilde{C}}_6$	[4,34,4]	File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	71
3	${\tilde{B}}_6$	h[4,34,4] [4,33,31,1]	File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	95 (32 new)
4	${\tilde{D}}_6$	q[4,34,4] [31,1,32,31,1]	File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png	41 (6 new)
5	${\tilde{E}}_6$	[32,2,2]	File:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	39

Regular and uniform tessellations include:

• ${\tilde{A}}_6$, 17 forms
  • Uniform 6-simplex honeycomb: {3^[7]} File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel branch.png
  • Uniform Cyclotruncated 6-simplex honeycomb: t_0,1{3^[7]} File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel branch 11.png
  • Uniform Omnitruncated 6-simplex honeycomb: t_{0,1,2,3,4,5,6,7}{3^[7]} File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel 3ab.pngFile:CDel nodes 11.pngFile:CDel 3ab.pngFile:CDel nodes 11.pngFile:CDel 3ab.pngFile:CDel branch 11.png
• ${\tilde{C}}_6$, [4,3⁴,4], 71 forms
  • Regular 6-cube honeycomb, represented by symbols {4,3⁴,4}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png
• ${\tilde{B}}_6$, [3^1,1,3³,4], 95 forms, 64 shared with ${\tilde{C}}_6$, 32 new
  • Uniform 6-demicube honeycomb, represented by symbols h{4,3⁴,4} = {3^1,1,3³,4}, File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png = File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png
• ${\tilde{D}}_6$, [3^1,1,3²,3^1,1], 41 unique ringed permutations, most shared with ${\tilde{B}}_6$ and ${\tilde{C}}_6$, and 6 are new. Coxeter calls the first one a quarter 6-cubic honeycomb.
  • File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h1.png
  • File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h1.png
  • File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h1.png
  • File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h1.png
  • File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 10lu.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h1.png
  • File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 10lu.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h1.png
• ${\tilde{E}}_6$: [3^2,2,2], 39 forms
  • Uniform 2₂₂ honeycomb: represented by symbols {3,3,3^2,2}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png
  • Uniform t₄(2₂₂) honeycomb: 4r{3,3,3^2,2}, File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 11.png
  • Uniform 0₂₂₂ honeycomb: {3^2,2,2}, File:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
  • Uniform t₂(0₂₂₂) honeycomb: 2r{3^2,2,2}, File:CDel nodes 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png

Prismatic groups

#	Coxeter group		Coxeter-Dynkin diagram
1	${\tilde{A}}_5$x${\tilde{I}}_1$	[3[6],2,∞]	File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
2	${\tilde{B}}_5$x${\tilde{I}}_1$	[4,3,31,1,2,∞]	File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
3	${\tilde{C}}_5$x${\tilde{I}}_1$	[4,33,4,2,∞]	File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
4	${\tilde{D}}_5$x${\tilde{I}}_1$	[31,1,3,31,1,2,∞]	File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
5	${\tilde{A}}_4$x${\tilde{I}}_1$x${\tilde{I}}_1$	[3[5],2,∞,2,∞,2,∞]	File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
6	${\tilde{B}}_4$x${\tilde{I}}_1$x${\tilde{I}}_1$	[4,3,31,1,2,∞,2,∞]	File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
7	${\tilde{C}}_4$x${\tilde{I}}_1$x${\tilde{I}}_1$	[4,3,3,4,2,∞,2,∞]	File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
8	${\tilde{D}}_4$x${\tilde{I}}_1$x${\tilde{I}}_1$	[31,1,1,1,2,∞,2,∞]	File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
9	${\tilde{F}}_4$x${\tilde{I}}_1$x${\tilde{I}}_1$	[3,4,3,3,2,∞,2,∞]	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
10	${\tilde{C}}_3$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$	[4,3,4,2,∞,2,∞,2,∞]	File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
11	${\tilde{B}}_3$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$	[4,31,1,2,∞,2,∞,2,∞]	File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
12	${\tilde{A}}_3$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$	[3[4],2,∞,2,∞,2,∞]	File:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
13	${\tilde{C}}_2$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$	[4,4,2,∞,2,∞,2,∞,2,∞]	File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
14	${\tilde{H}}_2$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$	[6,3,2,∞,2,∞,2,∞,2,∞]	File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
15	${\tilde{A}}_2$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$	[3[3],2,∞,2,∞,2,∞,2,∞]	File:CDel node.pngFile:CDel split1.pngFile:CDel branch.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
16	${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$x${\tilde{I}}_1$	[∞,2,∞,2,∞,2,∞,2,∞]	File:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png

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 3 paracompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams.

${\overline{P}}_6$ = [3,3[6]]: File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png

${\overline{Q}}_6$ = [31,1,3,32,1]: File:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

${\overline{S}}_6$ = [4,3,3,32,1]: File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 4a.pngFile:CDel nodea.png

Notes on the Wythoff construction for the uniform 7-polytopes

The reflective 7-dimensional uniform polytopes 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 polytopes 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.

Operation	Extended Schläfli symbol	Coxeter- Dynkin diagram	Description
Parent	t0{p,q,r,s,t,u}	File:CDel node 1.pngFile:CDel p.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	Any regular 7-polytope
Rectified	t1{p,q,r,s,t,u}	File:CDel node.pngFile:CDel p.pngFile:CDel node 1.pngFile:CDel q.pngFile:CDel node.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	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}	File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node 1.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	Birectification reduces cells to their duals.
Truncated	t0,1{p,q,r,s,t,u}	File:CDel node 1.pngFile:CDel p.pngFile:CDel node 1.pngFile:CDel q.pngFile:CDel node.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	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. File:Cube truncation sequence.svg
Bitruncated	t1,2{p,q,r,s,t,u}	File:CDel node.pngFile:CDel p.pngFile:CDel node 1.pngFile:CDel q.pngFile:CDel node 1.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	Bitrunction transforms cells to their dual truncation.
Tritruncated	t2,3{p,q,r,s,t,u}	File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node 1.pngFile:CDel r.pngFile:CDel node 1.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	Tritruncation transforms 4-faces to their dual truncation.
Cantellated	t0,2{p,q,r,s,t,u}	File:CDel node 1.pngFile:CDel p.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node 1.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	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. File:Cube cantellation sequence.svg
Bicantellated	t1,3{p,q,r,s,t,u}	File:CDel node.pngFile:CDel p.pngFile:CDel node 1.pngFile:CDel q.pngFile:CDel node.pngFile:CDel r.pngFile:CDel node 1.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	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.
Runcinated	t0,3{p,q,r,s,t,u}	File:CDel node 1.pngFile:CDel p.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node.pngFile:CDel r.pngFile:CDel node 1.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	Runcination reduces cells and creates new cells at the vertices and edges.
Biruncinated	t1,4{p,q,r,s,t,u}	File:CDel node.pngFile:CDel p.pngFile:CDel node 1.pngFile:CDel q.pngFile:CDel node.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node 1.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	Runcination reduces cells and creates new cells at the vertices and edges.
Stericated	t0,4{p,q,r,s,t,u}	File:CDel node 1.pngFile:CDel p.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node 1.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node.png	Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
Pentellated	t0,5{p,q,r,s,t,u}	File:CDel node 1.pngFile:CDel p.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node 1.pngFile:CDel u.pngFile:CDel node.png	Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps.
Hexicated	t0,6{p,q,r,s,t,u}	File:CDel node 1.pngFile:CDel p.pngFile:CDel node.pngFile:CDel q.pngFile:CDel node.pngFile:CDel r.pngFile:CDel node.pngFile:CDel s.pngFile:CDel node.pngFile:CDel t.pngFile:CDel node.pngFile:CDel u.pngFile:CDel node 1.png	Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. (expansion operation for 7-polytopes)
Omnitruncated	t0,1,2,3,4,5,6{p,q,r,s,t,u}	File:CDel node 1.pngFile:CDel p.pngFile:CDel node 1.pngFile:CDel q.pngFile:CDel node 1.pngFile:CDel r.pngFile:CDel node 1.pngFile:CDel s.pngFile:CDel node 1.pngFile:CDel t.pngFile:CDel node 1.pngFile:CDel u.pngFile:CDel node 1.png	All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied.

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
• H.S.M. Coxeter:
  • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Klitzing, Richard. "7D uniform polytopes (polyexa)".

External links

• Polytope names
• Polytopes of Various Dimensions
• Multi-dimensional Glossary