Cantic 7-cube

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Cantic 7-cube
Truncated 7-demicube
File:Truncated 7-demicube D7.svg
D7 Coxeter plane projection
Type uniform 7-polytope
Schläfli symbol t{3,34,1}
h2{4,3,3,3,3,3}
Coxeter diagram 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 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:CDel node h.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 3.pngFile:CDel node.png
6-faces 14 truncated 6-demicubes
64 truncated 6-simplexes
64 rectified 6-simplexes
5-faces 84 truncated 5-demicubes
448 truncated 5-simplexes
448 rectified 5-simplexes
448 5-simplexes
4-faces 280 truncated 16-cells
1344 truncated 5-cells
1344 rectified 5-cells
2688 5-cells
Cells 560 truncated tetrahedra
2240 truncated tetrahedra
2240 octahedra
6720 tetrahedra
Faces 2240 hexagons
2240 triangles
8960 triangles
Edges 672 segments
6720 segments
Vertices 1344
Vertex figure ( )v{ }x{3,3,3}
Coxeter groups D7, [34,1,1]
Properties convex

In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.

A uniform 7-polytope is vertex-transitive and constructed from uniform 6-polytope facets, and can be represented a coxeter diagram with ringed nodes representing active mirrors. A demihypercube is an alternation of a hypercube.

Its 3-dimensional analogue would be a truncated tetrahedron (truncated 3-demicube), and Coxeter diagram File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.png or File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png as a cantic cube.

Alternate names

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  • Truncated demihepteract
  • Truncated hemihepteract (acronym: thesa) (Jonathan Bowers)[1]

Cartesian coordinates

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The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6√2 are coordinate permutations:

(±1,±1,±3,±3,±3,±3,±3)

with an odd number of plus signs.

Images

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It can be visualized as a 2-dimensional orthogonal projections, for example the a D7 Coxeter plane, containing 12-gonal symmetry. Most visualizations in symmetric projections will contain overlapping vertices, so the colors of the vertices are changed based on how many vertices are at each projective position, here shown with red color for no overlaps.

orthographic projections
Coxeter
plane 		B7 D7 D6
Graph File:7-demicube t01 B7.svg File:7-demicube t01 D7.svg File:7-demicube t01 D6.svg
Dihedral
symmetry 		[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph File:7-demicube t01 D5.svg File:7-demicube t01 D4.svg File:7-demicube t01 D3.svg
Dihedral
symmetry 		[8] [6] [4]
Coxeter
plane 		A5 A3
Graph File:7-demicube t01 A5.svg File:7-demicube t01 A3.svg
Dihedral
symmetry 		[6] [4]
[edit | edit source]
Dimensional family of cantic n-cubes
n 3 4 5 6 7 8
Symmetry
[1+,4,3n-2] 		[1+,4,3]
= [3,3] 		[1+,4,32]
= [3,31,1] 		[1+,4,33]
= [3,32,1] 		[1+,4,34]
= [3,33,1] 		[1+,4,35]
= [3,34,1] 		[1+,4,36]
= [3,35,1]
Cantic
figure 		File:Cantic cube.png File:Schlegel half-solid truncated 16-cell.png File:Truncated 5-demicube D5.svg File:Truncated 6-demicube D6.svg File:Truncated 7-demicube D7.svg File:Truncated 8-demicube D8.svg
Coxeter File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.png 		File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.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.png
= File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.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.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 3.pngFile:CDel node.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 3.pngFile:CDel node.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 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.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 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.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 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Schläfli h2{4,3} h2{4,32} h2{4,33} h2{4,34} h2{4,35} h2{4,36}

There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:

D7 polytopes
File:7-demicube t0 D7.svg
t0(141) 		File:7-demicube t01 D7.svg
t0,1(141) 		File:7-demicube t02 D7.svg
t0,2(141) 		File:7-demicube t03 D7.svg
t0,3(141) 		File:7-demicube t04 D7.svg
t0,4(141) 		File:7-demicube t05 D7.svg
t0,5(141) 		File:7-demicube t012 D7.svg
t0,1,2(141) 		File:7-demicube t013 D7.svg
t0,1,3(141)
File:7-demicube t014 D7.svg
t0,1,4(141) 		File:7-demicube t015 D7.svg
t0,1,5(141) 		File:7-demicube t023 D7.svg
t0,2,3(141) 		File:7-demicube t024 D7.svg
t0,2,4(141) 		File:7-demicube t025 D7.svg
t0,2,5(141) 		File:7-demicube t034 D7.svg
t0,3,4(141) 		File:7-demicube t035 D7.svg
t0,3,5(141) 		File:7-demicube t045 D7.svg
t0,4,5(141)
File:7-demicube t0123 D7.svg
t0,1,2,3(141) 		File:7-demicube t0124 D7.svg
t0,1,2,4(141) 		File:7-demicube t0125 D7.svg
t0,1,2,5(141) 		File:7-demicube t0134 D7.svg
t0,1,3,4(141) 		File:7-demicube t0135 D7.svg
t0,1,3,5(141) 		File:7-demicube t0145 D7.svg
t0,1,4,5(141) 		File:7-demicube t0234 D7.svg
t0,2,3,4(141) 		File:7-demicube t0235 D7.svg
t0,2,3,5(141)
File:7-demicube t0245 D7.svg
t0,2,4,5(141) 		File:7-demicube t0345 D7.svg
t0,3,4,5(141) 		File:7-demicube t01234 D7.svg
t0,1,2,3,4(141) 		File:7-demicube t01235 D7.svg
t0,1,2,3,5(141) 		File:7-demicube t01245 D7.svg
t0,1,2,4,5(141) 		File:7-demicube t01345 D7.svg
t0,1,3,4,5(141) 		File:7-demicube t02345 D7.svg
t0,2,3,4,5(141) 		File:7-demicube t012345 D7.svg
t0,1,2,3,4,5(141)

Notes

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  1. ^ Klitzing, (x3x3o *b3o3o3o3o - thesa).

References

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  • H.S.M. Coxeter:
    • 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, wiley.com, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
      • (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 Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). x3x3o *b3o3o3o3o – thesa
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Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations
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