Hexic 7-cubes

7-demicube

Hexic 7-cube

Hexicantic 7-cube

Hexiruncic 7-cube

Hexiruncicantic 7-cube

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Hexisteric 7-cube

Hexistericantic 7-cube

Hexisteriruncic 7-cube

Hexisteriruncicantic 7-cube

Hexipentic 7-cube

Hexipenticantic 7-cube

Hexipentiruncic 7-cube

Hexipentiruncicantic 7-cube

Hexipentisteric 7-cube

Hexipentistericantic 7-cube

Hexipentisteriruncic 7-cube

Hexipentisteriruncicantic 7-cube

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Orthogonal projections in D7 Coxeter plane

In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms.

## Contents

* 1 Hexic 7-cube
* 1.1 Cartesian coordinates
* 1.2 Images
* 2 Hexicantic 7-cube
* 2.1 Images
* 3 Hexiruncic 7-cube
* 3.1 Images
* 4 Hexisteric 7-cube
* 4.1 Images
* 5 Hexipentic 7-cube
* 5.1 Images
* 6 Hexiruncicantic 7-cube
* 6.1 Images
* 7 Hexistericantic 7-cube
* 7.1 Images
* 8 Hexipenticantic 7-cube
* 8.1 Images
* 9 Hexisteriruncic 7-cube
* 9.1 Images
* 10 Hexipentiruncic 7-cube
* 10.1 Images
* 11 Hexipentisteric 7-cube
* 11.1 Images
* 12 Hexisteriruncicantic 7-cube
* 12.1 Images
* 13 Hexipentiruncicantic 7-cube
* 13.1 Images
* 14 Hexipentisteriruncic 7-cube
* 14.1 Images
* 15 Hexipentistericantic 7-cube
* 15.1 Images
* 16 Hexipentisteriruncicantic 7-cube
* 16.1 Images
* 17 Related polytopes
* 18 Notes
* 19 References
* 20 External links

## Hexic 7-cube

Hexic 7-cube

Type | uniform 7-polytope
Schläfli symbol | t0,5{3,34,1}
h6{4,35}
Coxeter-Dynkin diagram |

5-faces |
4-faces |
Cells |
Faces |
Edges | 4704
Vertices | 448
Vertex figure |
Coxeter groups | D7, [34,1,1]
Properties | convex

### Cartesian coordinates

The Cartesian coordinates for the vertices of a hexic 7-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

### Images

## Hexicantic 7-cube

### Images

## Hexiruncic 7-cube

### Images

## Hexisteric 7-cube

### Images

## Hexipentic 7-cube

### Images

## Hexiruncicantic 7-cube

### Images

## Hexistericantic 7-cube

### Images

## Hexipenticantic 7-cube

### Images

## Hexisteriruncic 7-cube

### Images

## Hexipentiruncic 7-cube

### Images

## Hexipentisteric 7-cube

### Images

## Hexisteriruncicantic 7-cube

### Images

## Hexipentiruncicantic 7-cube

### Images

## Hexipentisteriruncic 7-cube

### Images

## Hexipentistericantic 7-cube

### Images

## Hexipentisteriruncicantic 7-cube

### Images

## Related polytopes

This polytope is based on the 7-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC7 symmetry, and 32 are unique:

## Notes

## References

* 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, 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]
* Norman Johnson Uniform Polytopes, Manuscript (1991)
* N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
* Klitzing, Richard. "7D uniform polytopes (polyexa)". https://bendwavy.org/klitzing/dimensions/polyexa.htm.

## External links

* Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html.
* Polytopes of Various Dimensions
* Multi-dimensional Glossary

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* e

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 4-polytope | 5-cell | 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

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