| Regular 7-orthoplex (heptacross) |
|
|---|---|
Orthogonal projection inside Petrie polygon |
|
| Type | Regular 7-polytope |
| Family | orthoplex |
| Schläfli symbol | {35,4} {34,1,1} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 128 {35} |
| 5-faces | 448 {34} |
| 4-faces | 672 {33} |
| Cells | 560 {3,3} |
| Faces | 280 {3} |
| Edges | 84 |
| Vertices | 14 |
| Vertex figure | 6-orthoplex |
| Petrie polygon | tetradecagon |
| Coxeter groups | C7, [3,3,3,3,3,4] D7, [34,1,1] |
| Dual | 7-cube |
| Properties | convex |
In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.
It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {34,1,1} or Coxeter symbol 411.
Contents |
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph | |||
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract.
There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or [4,3,3,3,3,3] symmetry group, and a lower symmetry with two copies of 6-simplex facets, alternating, with the D7 or [34,1,1] symmetry group.
Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are
Every vertex pair is connected by an edge, except opposites.