7-orthoplex

Regular 7-orthoplex
(heptacross)

Orthogonal projection
inside Petrie polygon
TypeRegular 7-polytope
Familyorthoplex
Schläfli symbol {35,4}
{3,3,3,3,31,1}
Coxeter-Dynkin diagrams
6-faces128 {35}
5-faces448 {34}
4-faces672 {33}
Cells560 {3,3}
Faces280 {3}
Edges84
Vertices14
Vertex figure6-orthoplex
Petrie polygontetradecagon
Coxeter groupsC7, [3,3,3,3,3,4]
D7, [34,1,1]
Dual7-cube
Propertiesconvex

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 {3,3,3,3,31,1} or Coxeter symbol 411.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract.

Alternate names

Images

orthographic projections
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]

Construction

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 half symmetry with two copies of 6-simplex facets, alternating, with the D7 or [34,1,1] 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
regular 7-orthoplex {3,3,3,3,31,1} [3,3,3,3,31,1]322560
7-fusil 7{} [26]128

Cartesian coordinates

Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are

(±1,0,0,0,0,0,0), (0,±1,0,0,0,0,0), (0,0,±1,0,0,0,0), (0,0,0,±1,0,0,0), (0,0,0,0,±1,0,0), (0,0,0,0,0,±1,0), (0,0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.

See also

References

External links

This article is issued from Wikipedia - version of the Tuesday, January 27, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.