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 9-

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, ...

9-polytope In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope Ridge (geometry), ridge being shared by exactly two 8-polytope Facet (mathematics), facets. A uniform 9-polytope ...

. It has 10 vertices, 45

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, 120 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 ...

, 210 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 ...

, 252

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 ...

4-faces, 210

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-faces, 120

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 ...

6-faces, 45

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 ...

7-faces, and 10

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 angl ...

8-faces. Its dihedral angle is cos⁻¹(1/9), or approximately 83.62°. It can also be called a decayotton, or deca-9-tope, as a 10- facetted polytope in 9-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 ...

''decayotton'' is derived from ''deca'' for ten 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 yotta (a variation of "oct" for eight), having 8-dimensional facets, and ''-on''.
Jonathan Bowers gives it acronym day.

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 decayotton having edge length 2 are: :

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

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

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

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

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

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

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

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

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

More simply, the vertices of the ''9-simplex'' can be positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,0,1). These are the vertices of one

Facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...

of the 10-orthoplex.

Images

References

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

External links

*
Polytopes of Various Dimensions

{{Polytopes 9-polytopes