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Graphs of three regular and three uniform 7-polytopes.
Complete graph K8.svg
7-simplex
(Octaexon)
Cross graph 7.png
7-orthoplex, 411
(Heptacross)
Hepteract ortho petrie.svg
7-cube
(Hepteract)
Demihepteract ortho petrie.svg
7-demicube 141
(Demihepteract)
E7 graph.svg
321 polytope
Gosset 2 31 polytope.svg
231 polytope
Gosset 1 32 petrie.svg
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.

Contents

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] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
2 B7 [4,35] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
3 D7 [34,1,1] CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png
4 E7 [33,2,1] CD dot.pngCD 3.pngCD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png

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

  1. Simplex A7 family: [36] - CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
    • 71 uniform 7-polytopes as permutations of rings in the group diagram, including one regular:
      1. {36} - 7-simplex or octaexon, CDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
  2. Hypercube/orthoplex B7 family: [4,35] - CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
    • 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 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
      2. {35,4} - 7-orthoplex or heptacross CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.png
  3. Demihypercube D7 family: [34,1,1] - CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png
    • 95 uniform 7-polytope as permutations of rings in the group diagram, including:
      1. {31,4,1} - 7-demicube or demihepteract CD ring.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png; also as h{4,35} CDW hole.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
      2. {34,1,1} - heptacross CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD ring.png
  4. Semiregular E7 family:[33,2,1] - CD dot.pngCD 3.pngCD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png
    • 127 uniform 7-polytopes as permutations of rings in the group diagram, including:
      1. {33,2,1} - Thorold Gosset's semiregular 321, CD dot.pngCD 3.pngCD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD ring.png
      2. {31,3,2} - the uniform 132, CD dot.pngCD 3.pngCD dot.pngCD 3.pngCD downbranch-01.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png
      3. {32,3,1} - the uniform 231, CD ring.pngCD 3.pngCD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png

Uniform prismatic forms

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

# Coxeter group Coxeter-Dynkin diagram
1 A6×A1 [35] × [ ] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
2 B6×A1 [4,34] × [ ] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
3 D6×A1 [33,1,1] × [ ] CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 2.pngCD dot.png
4 E6×A1 [32,2,1] × [ ] CD dot.pngCD 3.pngCD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 2.pngCD dot.png
5 A4×I2(p)×A1 [3,3,3] × [p] × [ ] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.png
6 B4×I2(p)×A1 [4,3,3] × [p] × [ ] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.png
7 F4×I2(p)×A1 [3,4,3] × [p] × [ ] CDW dot.pngCDW 3.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.png
8 H4×I2(p)×A1 [5,3,3] × [p] × [ ] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.png
9 D4×I2(p)×A1 [31,1,1] × [p] × [ ] CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.png
10 A3×A3×A1 [3,3] × [3,3] × [ ] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
11 A3×B3×A1 [3,3] × [4,3] × [ ] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
12 A3×H3×A1 [3,3] × [5,3] × [ ] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
13 B3×B3×A1 [4,3] × [4,3] × [ ] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
14 B3×H3×A1 [4,3] × [5,3] × [ ] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
15 H3×A3×A1 [5,3] × [5,3] × [ ] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
16 I2(p)×I2(q)×I2(r)×A1 [p] × [q] × [r] × [ ] CDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW q.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW r.pngCDW dot.pngCDW 2.pngCDW dot.png

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] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.png
2 B5×I2(p) [4,3,3] × [p] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.png
3 D5×I2(p) [32,1,1] × [p] CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.pngCD 2.pngCD dot.pngCD p.pngCD dot.png
4 A4×A3 [3,3,3] × [3,3] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
5 A4×B3 [3,3,3] × [4,3] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.png
6 A4×H3 [3,3,3] × [5,3] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.png
7 B4×A3 [4,3,3] × [3,3] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
8 B4×B3 [4,3,3] × [4,3] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.png
9 B4×H3 [4,3,3] × [5,3] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.png
10 H4×A3 [5,3,3] × [3,3] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
11 H4×B3 [5,3,3] × [4,3] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.png
12 H4×H3 [5,3,3] × [5,3] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.png
13 F4×A3 [3,4,3] × [3,3] CDW dot.pngCDW 3.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
14 F4×B3 [3,4,3] × [4,3] CDW dot.pngCDW 3.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.png
15 F4×H3 [3,4,3] × [5,3] CDW dot.pngCDW 3.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.png
16 D4×A3 [31,1,1] × [3,3] CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCDW 2.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
17 D4×B3 [31,1,1] × [4,3] CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCDW 2.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.png
18 D4×H3 [31,1,1] × [5,3] CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCDW 2.pngCDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.png

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] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW q.pngCDW dot.png
2 B3×I2(p)×I2(q) [4,3] × [p] × [q] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW q.pngCDW dot.png
3 H3×I2(p)×I2(q) [5,3] × [p] × [q] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW q.pngCDW dot.png

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]] CD downbranch-00.pngCD downbranch-33.pngCD downbranch-open.pngCD downbranch-33.pngCD righttriangleopen 000.png
2 B~6 [4,34,4] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
3 C~6 h[4,34,4]
[4,33,31,1]
CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 4.pngCD dot.png
4 D~6 q[4,34,4]
[31,1,32,31,1]
CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png
5 E~6 [32,2,2] CD dot.pngCD 3.pngCD dot.pngCD 3.pngCD downbranch-00.pngCD downbranch-33.pngCD downbranch-open.pngCD 3.pngCD dot.png

Regular and uniform tessellations include:

See also

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]

External links

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