Regular 7-orthoplex (heptacross) | |
---|---|
Orthogonal projection inside Petrie polygon | |
Type | Regular 7-polytope |
Family | orthoplex |
Schläfli symbol | {3^{5},4} {3,3,3,3,3^{1,1}} |
Coxeter-Dynkin diagrams | |
6-faces | 128 {3^{5}} |
5-faces | 448 {3^{4}} |
4-faces | 672 {3^{3}} |
Cells | 560 {3,3} |
Faces | 280 {3} |
Edges | 84 |
Vertices | 14 |
Vertex figure | 6-orthoplex |
Petrie polygon | tetradecagon |
Coxeter groups | C_{7}, [3,3,3,3,3,4] D_{7}, [3^{4,1,1}] |
Dual | 7-cube |
Properties | convex, Hanner polytope |
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 {3^{5},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3^{1,1}} or Coxeter symbol 4_{11}.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract.
This configuration matrix represents the 7-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}
Coxeter plane | B_{7} / A_{6} | B_{6} / D_{7} | B_{5} / D_{6} / A_{4} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B_{4} / D_{5} | B_{3} / D_{4} / A_{2} | B_{2} / D_{3} |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A_{5} | A_{3} | |
Graph | |||
Dihedral symmetry | [6] | [4] |
There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C_{7} or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D_{7} or [3^{4,1,1}] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.
Name | Coxeter diagram | Schläfli symbol | Symmetry | Order | Vertex figure |
---|---|---|---|---|---|
regular 7-orthoplex | {3,3,3,3,3,4} | [3,3,3,3,3,4] | 645120 | ||
Quasiregular 7-orthoplex | {3,3,3,3,3^{1,1}} | [3,3,3,3,3^{1,1}] | 322560 | ||
7-fusil | 7{} | [2^{6}] | 128 |
Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are
Every vertex pair is connected by an edge, except opposites.