In nine-dimensional

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 rectified 9-simplex is a convex

uniform 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 being shared by exactly two 8-polytope facets. A uniform 9-polytope is one which is vertex-transitive, an ...

, being a

rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...

of the regular

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 ...

. These polytopes are part of a family of 271

s with A₉ symmetry. There are unique 4 degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex are located in the triangular face centers of the 9-simplex. Vertices of the trirectified 9-simplex are located in the

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 ...

cell centers of the 9-simplex. Vertices of the quadrirectified 9-simplex are located in the

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 ...

centers of the 9-simplex.

Rectified 9-simplex

The rectified 9-simplex is the

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...

of the

10-demicube In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it ...

Alternate names

* Rectified decayotton (reday) (Jonathan Bowers)

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 the ''rectified 9-simplex'' can be most simply positioned in 10-space as

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...

s of (0,0,0,0,0,0,0,0,1,1). This construction is based on

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 ...

of the

rectified 10-orthoplex In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the e ...

Images

Birectified 9-simplex

This polytope is the

for the 1₆₂ honeycomb. Its 120 vertices represent the

kissing number In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...

of the related hyperbolic 10-dimensional

sphere packing In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three- dimensional Euclidean space. However, sphere pack ...

Alternate names

* Birectified decayotton (breday) (Jonathan Bowers)

Coordinates

The

s of the vertices of the ''birectified 9-simplex'' can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,1,1,1). This construction is based on

of the

birectified 10-orthoplex In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the e ...

Images

Trirectified 9-simplex

Alternate names

* Trirectified decayotton (treday) (Jonathan Bowers)

Coordinates

The

s of the vertices of the ''trirectified 9-simplex'' can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,1,1,1,1). This construction is based on

of the

trirectified 10-orthoplex In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the e ...

Images

Quadrirectified 9-simplex

Alternate names

* Quadrirectified decayotton * Icosayotton (icoy) (Jonathan Bowers)Klitzing, (o3o3o3o3x3o3o3o3o - icoy)

Coordinates

The

s of the vertices of the ''quadrirectified 9-simplex'' can be most simply positioned in 10-space as permutations of (0,0,0,0,0,1,1,1,1,1). This construction is based on

of the

quadrirectified 10-orthoplex In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the ed ...

Images

Notes

References

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

: ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 ** Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) * o3x3o3o3o3o3o3o3o - reday, o3o3x3o3o3o3o3o3o - breday, o3o3o3x3o3o3o3o3o - treday, o3o3o3o3x3o3o3o3o - icoy

External links

Polytopes of Various Dimensions

{{Polytopes 9-polytopes