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 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 ...

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 cons ...

. 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 b ...

s, 165 triangle faces, 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 is ...

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-s ...

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°. Alt ...

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 is c ...

8-faces, and 11

9-simplex In geometry, a 9-simplex is a self-dual Regular polytope, regular 9-polytope. It has 10 vertex (geometry), vertices, 45 Edge (geometry), edges, 120 triangle Face (geometry), faces, 210 tetrahedral Cell (mathematics), cells, 252 5-cell 4-faces, 210 ...

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 ''hendecaronnon'' is derived from ''hendeca'' for 11 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 -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 i ...

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 facets of the

11-orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahe ...

Images

Related polytopes

The 2-skeleton of the 10-simplex is topologically related to the 11-cell 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