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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a 6-
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 ...
6-polytope. It has 7 vertices, 21
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 b ...
s, 35 triangle faces, 35
tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
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 ...
, 21
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 is ...
4-faces, and 7
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 ...
5-faces. Its
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
is cos−1(1/6), or approximately 80.41°.


Alternate names

It can also be called a heptapeton, or hepta-6-tope, as a 7- facetted polytope in 6-dimensions. The name ''heptapeton'' is derived from ''hepta'' for seven facets in
Greek Greek may refer to: Greece 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 ...
and ''-peta'' for having five-dimensional facets, and ''-on''. Jonathan Bowers gives a heptapeton the acronym hop.


As a configuration

This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-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\begin7 & 6 & 15 & 20 & 15 & 6 \\ 2 & 21 & 5 & 10 & 10 & 5 \\ 3 & 3 & 35 & 4 & 6 & 4 \\ 4 & 6 & 4 & 35 & 3 & 3 \\ 5 & 10 & 10 & 5 & 21 & 2 \\ 6 & 15 & 20 & 15 & 6 & 7 \end\end


Coordinates

The
Cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
s for an origin-centered regular heptapeton having edge length 2 are: :\left(\sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \pm1\right) :\left(\sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0\right) :\left(\sqrt,\ \sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0\right) :\left(\sqrt,\ \sqrt,\ -2\sqrt,\ 0,\ 0,\ 0\right) :\left(\sqrt,\ -\sqrt,\ 0,\ 0,\ 0,\ 0\right) :\left(-\sqrt,\ 0,\ 0,\ 0,\ 0,\ 0\right) The vertices of the ''6-simplex'' can be more simply positioned in 7-space as permutations of: : (0,0,0,0,0,0,1) This construction is based on facets of the 7-orthoplex.


Images


Related uniform 6-polytopes

The regular 6-simplex is one of 35 uniform 6-polytopes based on the ,3,3,3,3
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
, all shown here in A6
Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal ...
s.


Notes


References

* Coxeter, H.S.M.: ** ** *** (Paper 22) *** (Paper 23) *** (Paper 24) * * **


External links

*
Polytopes of Various Dimensions


{{Polytopes 6-polytopes