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
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
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.
Coordinates
The
Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are:
:
:
:
:
:
:
:
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 A
7 symmetry.
Notes
External links
*
Polytopes of Various Dimensions
{{Polytopes
7-polytopes