# Uniform 7-polytope: Wikis

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 7-simplex (Octaexon) 7-orthoplex, 411 (Heptacross) 7-cube (Hepteract) 7-demicube 141 (Demihepteract) 321 polytope 231 polytope 131 polytope

In geometry, a seven-dimensional polytope, or 7-polytope, is a polytope in 7-dimensional space. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

A proposed name for 7-polytope is polyexon (plural: polyexa), created from poly-, exa- (short for hexa, meaning six) and -on.

## 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.

## Euler characteristic

The Euler characteristic for 7-polytopes that are topological 6-spheres (including all convex 7-polytopes) is two. χ=V-E+F-C+f4-f5+f6=2.

For the 3 convex regular 7-polytopes, their elements are:

Name Schläfli
symbol
Vertices Edges Faces Cells 4-faces 5-faces 6-faces χ
7-simplex {3,3,3,3,3,3} 8 28 56 70 56 28 8 2
7-cube {4,3,3,3,3,3} 128 448 672 560 280 84 14 2
7-orthoplex {3,3,3,3,3,4} 14 84 280 560 672 448 128 2

For 4 convex uniform 7-polytopes, their elements are:

Name Schläfli
symbol
Vertices Edges Faces Cells 4-faces 5-faces 6-faces χ
7-demicube {31,4,1} 64 672 2240 2800 1624 532 78 2
321 polytope {33,2,1} 56 756 4032 10080 12096 6048 702 2
231 polytope {32,3,1} 126 2016 10080 20160 16128 4788 632 2
132 polytope {31,3,2} 576 10080 40320 50400 23688 4284 182 2

## 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 Coxeter-Dynkin diagram
1 A7 [36]
2 B7 [4,35]
3 D7 [34,1,1]
4 E7 [33,2,1]

Selected regular and uniform 7-polytopes from each family include:

1. Simplex A7 family: [36] -
• 71 uniform 7-polytopes as permutations of rings in the group diagram, including one regular:
1. {36} - 7-simplex or octaexon,
2. Hypercube/orthoplex B7 family: [4,35] -
• 127 uniform 7-polytopes as permutations of rings in the group diagram, including two regular ones. There's also an alternated regular one.
1. {4,35} - 7-cube or hepteract
2. {35,4} - 7-orthoplex or heptacross
3. Demihypercube D7 family: [34,1,1] -
• 95 uniform 7-polytope as permutations of rings in the group diagram, including:
1. {31,4,1} - 7-demicube or demihepteract ; also as h{4,35}
2. {34,1,1} - heptacross
4. Semiregular E7 family:[33,2,1] -
• 127 uniform 7-polytopes as permutations of rings in the group diagram, including:
1. {33,2,1} - Thorold Gosset's semiregular 321,
2. {31,3,2} - the uniform 132,
3. {32,3,1} - the uniform 231,

## Uniform prismatic forms

There are 16 uniform prismatic families based on the uniform 6-polytopes.

# Coxeter group Coxeter-Dynkin diagram
1 A6×A1 [35] × [ ]
2 B6×A1 [4,34] × [ ]
3 D6×A1 [33,1,1] × [ ]
4 E6×A1 [32,2,1] × [ ]
5 A4×I2(p)×A1 [3,3,3] × [p] × [ ]
6 B4×I2(p)×A1 [4,3,3] × [p] × [ ]
7 F4×I2(p)×A1 [3,4,3] × [p] × [ ]
8 H4×I2(p)×A1 [5,3,3] × [p] × [ ]
9 D4×I2(p)×A1 [31,1,1] × [p] × [ ]
10 A3×A3×A1 [3,3] × [3,3] × [ ]
11 A3×B3×A1 [3,3] × [4,3] × [ ]
12 A3×H3×A1 [3,3] × [5,3] × [ ]
13 B3×B3×A1 [4,3] × [4,3] × [ ]
14 B3×H3×A1 [4,3] × [5,3] × [ ]
15 H3×A3×A1 [5,3] × [5,3] × [ ]
16 I2(p)×I2(q)×I2(r)×A1 [p] × [q] × [r] × [ ]

## Uniform duoprismatic forms

There are 18 uniform duoprismatic forms based on Cartesian products of lower dimensional uniform polytopes.

# Coxeter group Coxeter-Dynkin diagram
1 A5×I2(p) [3,3,3] × [p]
2 B5×I2(p) [4,3,3] × [p]
3 D5×I2(p) [32,1,1] × [p]
4 A4×A3 [3,3,3] × [3,3]
5 A4×B3 [3,3,3] × [4,3]
6 A4×H3 [3,3,3] × [5,3]
7 B4×A3 [4,3,3] × [3,3]
8 B4×B3 [4,3,3] × [4,3]
9 B4×H3 [4,3,3] × [5,3]
10 H4×A3 [5,3,3] × [3,3]
11 H4×B3 [5,3,3] × [4,3]
12 H4×H3 [5,3,3] × [5,3]
13 F4×A3 [3,4,3] × [3,3]
14 F4×B3 [3,4,3] × [4,3]
15 F4×H3 [3,4,3] × [5,3]
16 D4×A3 [31,1,1] × [3,3]
17 D4×B3 [31,1,1] × [4,3]
18 D4×H3 [31,1,1] × [5,3]

## Uniform triprismatic forms

There are 3 uniform triprismatic families based on Cartesian products of uniform polyhedrons and two regular polygons.

# Coxeter group Coxeter-Dynkin diagram
1 A3×I2(p)×I2(q) [3,3] × [p] × [q]
2 B3×I2(p)×I2(q) [4,3] × [p] × [q]
3 H3×I2(p)×I2(q) [5,3] × [p] × [q]

## Regular and uniform honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 6-space:

# Coxeter group Coxeter-Dynkin diagram
1 A~6 [3[7]]
2 B~6 [4,34,4]
3 C~6 h[4,34,4]
[4,33,31,1]
4 D~6 q[4,34,4]
[31,1,32,31,1]
5 E~6 [32,2,2]

Regular and uniform tessellations include:

## 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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (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
• Richard Klitzing 7D quasiregulars, prisms & duoprisms, duoprismatic prisms, triprisms & triprismatic prisms [2]