| Orthogonal projections in D7 Coxeter plane |
|---|
| 7-demicube (half 7-cube, h{4,35}) | Pentic 7-cube h5{4,35} | Penticantic 7-cube h2,5{4,35} |
| Pentiruncic 7-cube h3,5{4,35} | Pentiruncicantic 7-cube h2,3,5{4,35} | Pentisteric 7-cube h4,5{4,35} |
| Pentistericantic 7-cube h2,4,5{4,35} | Pentisteriruncic 7-cube h3,4,5{4,35} | Penticsteriruncicantic 7-cube h2,3,4,5{4,35} |
In seven-dimensional geometry, a pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms.
Pentic 7-cube
| Pentic 7-cube |
|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,4{3,34,1} h5{4,35} |
| Coxeter-Dynkin diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 1344 |
| Vertex figure | |
| Coxeter groups | D7, [34,1,1] |
| Properties | convex |
Alternate names
- Small cellated demihepteract (acronym: sochesa)
Cartesian coordinates
The Cartesian coordinates for the vertices of a pentic 7-cube centered at the origin are coordinate permutations:
(±1,±1,±1,±1,±1,±3,±3)
with an odd number of plus signs.
Images
Orthographic projections| Coxeter plane | B7 | D7 | D6 |
|---|
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
Related polytopes
| Dimensional family of pentic n-cubes |
|---|
| n | 6 | 7 | 8 |
| [1+,4,3n-2] = [3,3n-3,1] | [1+,4,34] = [3,33,1] | [1+,4,35] = [3,34,1] | [1+,4,36] = [3,35,1] |
| Cantic figure | | | |
| Coxeter | = | = | = |
| Schläfli | h5{4,34} | h5{4,35} | h5{4,36} |
Penticantic 7-cube
Alternate names
- Cellitruncated demihepteract (acronym: cothesa)
Images
Orthographic projections| Coxeter plane | B7 | D7 | D6 |
|---|
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
Pentiruncic 7-cube
Alternate names
- Cellirhombated demihepteract (acronym: crohesa)
Images
Orthographic projections| Coxeter plane | B7 | D7 | D6 |
|---|
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
Pentiruncicantic 7-cube
Alternate names
- Celligreatorhombated demihepteract (acronym: cagrohesa)
Images
Orthographic projections| Coxeter plane | B7 | D7 | D6 |
|---|
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
Pentisteric 7-cube
Alternate names
- Celliprismated demihepteract (acronym: caphesa)
Images
Orthographic projections| Coxeter plane | B7 | D7 | D6 |
|---|
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
Pentistericantic 7-cube
Alternate names
- Celliprismatotruncated demihepteract (acronym: capthesa)
Images
Orthographic projections| Coxeter plane | B7 | D7 | D6 |
|---|
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
Pentisteriruncic 7-cube
Alternate names
- Celliprismatorhombated demihepteract (acronym: coprahesa)
Images
Orthographic projections| Coxeter plane | B7 | D7 | D6 |
|---|
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
Pentisteriruncicantic 7-cube
Alternate names
- Great cellated demihepteract (acronym: gochesa)
Images
Orthographic projections| Coxeter plane | B7 | D7 | D6 |
|---|
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
Related polytopes
These polytopes are based on the 7-demicube, a member 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:
| D7 polytopes |
|---|
| t0(141) | t0,1(141) | t0,2(141) | t0,3(141) | t0,4(141) | t0,5(141) | t0,1,2(141) | t0,1,3(141) |
| t0,1,4(141) | t0,1,5(141) | t0,2,3(141) | t0,2,4(141) | t0,2,5(141) | t0,3,4(141) | t0,3,5(141) | t0,4,5(141) |
| t0,1,2,3(141) | t0,1,2,4(141) | t0,1,2,5(141) | t0,1,3,4(141) | t0,1,3,5(141) | t0,1,4,5(141) | t0,2,3,4(141) | t0,2,3,5(141) |
| t0,2,4,5(141) | t0,3,4,5(141) | t0,1,2,3,4(141) | t0,1,2,3,5(141) | t0,1,2,4,5(141) | t0,1,3,4,5(141) | t0,2,3,4,5(141) | t0,1,2,3,4,5(141) |
Notes
- 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 Ivić Weiss, Wiley-Interscience Publication, 1995, , ISBN 978-0-471-01003-6 (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. .
External links