In 7-dimensional

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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 Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

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 who ...

. It has 8 vertices, 28

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

s, 56 triangle

faces The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...

, 70 tetrahedral

cells Cell most often refers to: * Cell (biology), the functional basic unit of life * Cellphone, a phone connected to a cellular network * Clandestine cell, a penetration-resistant form of a secret or outlawed organization * Electrochemical cell, a d ...

, 56

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

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 ...

6-faces, and 8

6-simplex In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. A ...

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 person ...

''octaexon'' is derived from ''octa'' for eight facets in

Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...

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

Symmetry

There are many lower symmetry constructions of the 7-simplex. Some are expressed as join partitions of two or more lower simplexes. The symmetry order of each join is the product of the symmetry order of the elements, and raised further if identical elements can be interchanged.

Coordinates

The

Cartesian coordinate In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

s 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 In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 Vertex (geometry), vertices, 112 Edge (geometry), edges, 448 triangle Face (geometry), faces, 1120 tetrahedron Cell (mathematics), cells, 1792 5-cell ''4-faces'', 179 ...

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-polytope In seven-dimensional space, seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope Ridge (geometry), ridge being shared by exactly two 6-polytope Facet (mathematics), facets. A uniform 7-polytope is ...

s with A₇ symmetry.

Notes

External links

*
Polytopes of Various Dimensions

{{Polytopes 7-polytopes