In six-dimensional
geometry, a rectified 6-orthoplex is a convex
uniform 6-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
6-orthoplex.
There are unique 6 degrees of rectifications, the zeroth being the
6-orthoplex, and the 6th and last being the
6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.
Rectified 6-orthoplex
The ''rectified 6-orthoplex'' is the
vertex figure for the
demihexeractic honeycomb
The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.
It is composed of two different types of facet ...
.
: or
Alternate names
* rectified hexacross
* rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)
Construction
There are two
Coxeter groups associated with the ''rectified hexacross'', one with the C
6 or
,3,3,3,3Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D
6 or
3,1,1Coxeter 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 hexacross, centered at the origin, edge length
are all permutations of:
: (±1,±1,0,0,0,0)
Images
Root vectors
The 60 vertices represent the root vectors of the
simple Lie group D
6. 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 15 vertices
rectified 5-simplices cells on opposite sides, and 30 vertices of an
expanded 5-simplex
In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.
There are six unique sterications of the 5-simplex, including permutations of truncations, ...
passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B
6 and C
6 simple Lie groups.
The 60 roots of D
6 can be
geometrically folded into H
3 (
Icosahedral symmetry), as to , creating 2 copies of 30-vertex
icosidodecahedra, with the
Golden ratio between their radii:
Icosidodecahedron from D6
John Baez, January 1, 2015
Birectified 6-orthoplex
The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb
The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space.
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.
Constructions
There ar ...
.
Alternate names
* birectified hexacross
* birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)
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 hexacross, centered at the origin, edge length are all permutations of:
: (±1,±1,±1,0,0,0)
Images
It can also be projected into 3D-dimensions as --> , a dodecahedron envelope.
Related polytopes
These polytopes are a part a family of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
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.
* o3x3o3o3o4o - rag, o3o3x3o3o4o - brag
External links
Polytopes of Various Dimensions
{{polytopes
6-polytopes