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 10-

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

10-polytope In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. A uniform 10-polytope is one which is vertex-transitive, and construct ...

. It has 11 vertices, 55

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 ...

s, 165 triangle

faces The face is the front of an animal's 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 affe ...

, 330 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 ...

, 462

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 ...

4-faces, 462

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 ...

5-faces, 330

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°. Alte ...

6-faces, 165

7-simplex In 7-dimensional geometry, a 7- simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1 ...

7-faces, 55

8-simplex In geometry, an 8- simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle i ...

8-faces, and 11

9-simplex In geometry, a 9- simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-fa ...

9-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(1/10), or approximately 84.26°. It can also be called a hendecaronnon, or hendeca-10-tope, as an 11- facetted polytope in 10-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 persona ...

''hendecaronnon'' is derived from ''hendeca'' for 11

facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

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 -ronn (variation of ennea for nine), having 9-dimensional facets, and ''-on''.

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 in ...

s of the vertices of an origin-centered regular 10-simplex having edge length 2 are: :

\left(\sqrt,\ \sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \pm1\right)

\left(\sqrt,\ \sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0\right)

\left(\sqrt,\ \sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0\right)

\left(\sqrt,\ \sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0,\ 0,\ 0\right)

\left(\sqrt,\ \sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0,\ 0,\ 0\right)

\left(\sqrt,\ \sqrt,\ 1/6,\ \sqrt,\ -\sqrt,\ 0,\ 0,\ 0,\ 0,\ 0\right)

\left(\sqrt,\ \sqrt,\ 1/6,\ -\sqrt,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

\left(\sqrt,\ \sqrt,\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

\left(\sqrt,\ -3\sqrt,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

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

More simply, the vertices of the ''10-simplex'' can be positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,0,1). This construction is based on

of the 11-orthoplex.

Images

Related polytopes

The 2-skeleton of the 10-simplex is topologically related to the

11-cell In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...

abstract regular polychoron which has the same 11 vertices, 55 edges, but only 1/3 the faces (55).

References

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

External links

*
Polytopes of Various Dimensions

{{Polytopes 10-polytopes