In seven-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 7-orthoplex is a convex uniform 7-polytope, 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 7-orthoplex. There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex.

Rectified 7-orthoplex

The ''rectified 7-orthoplex'' 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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

for the

demihepteractic honeycomb The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb. It is composed of two different types of f ...

. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb. : or

Alternate names

* rectified heptacross * rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon

Images

Construction

There are two Coxeter groups associated with the ''rectified heptacross'', one with the C₇ or ,3,3,3,3,3Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D₇ or ^4,1,1Coxeter group.

Cartesian coordinates

Cartesian coordinates 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 t ...

for the vertices of a rectified heptacross, centered at the origin, edge length

\sqrt\

are all permutations of: : (±1,±1,0,0,0,0,0)

Root vectors

Its 84 vertices represent the root vectors of the simple Lie group D₇. The vertices can be seen in 3

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

s, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an

expanded 6-simplex In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations ...

passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B₇ and C₇ simple Lie groups.

Birectified 7-orthoplex

Alternate names

* Birectified heptacross * Birectified hecatonicosoctaexon (Acronym barz) (Jonathan Bowers) - birectified 128-faceted polyexonKlitzing, (o3o3x3o3o3o4o - barz)

Images

Cartesian coordinates

for the vertices of a birectified 7-orthoplex, centered at the origin, edge length

\sqrt\

are all permutations of: : (±1,±1,±1,0,0,0,0)

Trirectified 7-orthoplex

trirectified 7-orthoplex In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a Rectification (geometry), rectification of the regular 7-orthoplex. There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and ...

is the same as a

trirectified 7-cube In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube. There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Ve ...

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. * o3x3o3o3o3o4o - rez, o3o3x3o3o3o4o - barz

External links

Polytopes of Various Dimensions

{{polytopes 7-polytopes