In 7-dimensional geometry, a 7-

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

is a self-dual regular

7-polytope In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose f ...

. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral

cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...

, 56

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...

5-faces, 28

5-simplex In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The 5-s ...

6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos⁻¹(1/7), or approximately 81.79°.

Alternate names

It can also be called an octaexon, or octa-7-tope, as an 8- facetted polytope in 7-dimensions. The

name A name is a term used for identification by an external observer. They can identify a class or category of things, or a single thing, either uniquely, or within a given context. The entity identified by a name is called its referent. A personal ...

''octaexon'' is derived from ''octa'' for eight facets in Greek and ''-ex'' for having six-dimensional facets, and ''-on''. Jonathan Bowers gives an octaexon the acronym oca.

As a configuration

This configuration matrix represents the 7-simplex. 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-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.

\begin\begin8 & 7 & 21 & 35 & 35 & 21 & 7 \\ 2 & 28 & 6 & 15 & 20 & 15 & 6 \\ 3 & 3 & 56 & 5 & 10 & 10 & 5 \\ 4 & 6 & 4 & 70 & 4 & 6 & 4 \\ 5 & 10 & 10 & 5 & 56 & 3 & 3 \\ 6 & 15 & 20 & 15 & 6 & 28 & 2 \\ 7 & 21 & 35 & 35 & 21 & 7 & 8 \end\end

Coordinates

The Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are: :

\left(\sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \pm1\right)

\left(\sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0\right)

\left(\sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0\right)

\left(\sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0,\ 0,\ 0\right)

\left(\sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0,\ 0,\ 0\right)

\left(\sqrt,\ -\sqrt,\ 0,\ 0,\ 0,\ 0,\ 0\right)

\left(-\sqrt,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

More simply, the vertices of the ''7-simplex'' can be positioned in 8-space as permutations of (0,0,0,0,0,0,0,1). This construction is based on facets of the 8-orthoplex.

Images

Orthographic projections

Related polytopes

This polytope is a facet in the uniform tessellation 3₃₁ with Coxeter-Dynkin diagram: : This polytope is one of 71 uniform 7-polytopes with A₇ symmetry.

Notes

External links

*
Polytopes of Various Dimensions

{{Polytopes 7-polytopes