Rectified 6-orthoplex
   HOME

TheInfoList



OR:

In six-dimensional
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 rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular
6-orthoplex In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 Vertex (geometry), vertices, 60 Edge (geometry), edges, 160 triangle Face (geometry), faces, 240 tetrahedron Cell (mathematics), cells, 192 5-cell ''4-faces'', and 64 ...
. There are unique 6 degrees of rectifications, the zeroth being the
6-orthoplex In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 Vertex (geometry), vertices, 60 Edge (geometry), edges, 160 triangle Face (geometry), faces, 240 tetrahedron Cell (mathematics), cells, 192 5-cell ''4-faces'', and 64 ...
, and the 6th and last being the
6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It ...
. 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 In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
for the demihexeractic honeycomb. : or


Alternate names

* rectified hexacross * rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)


Construction

There are two
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
s associated with the ''rectified hexacross'', one with the C6 or ,3,3,3,3Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or 3,1,1Coxeter group.


Cartesian coordinates

Cartesian coordinates 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 ...
for the vertices of a rectified hexacross, centered at the origin, edge length \sqrt\ 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 In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
D6. The vertices can be seen in 3
hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
s, with the 15 vertices rectified 5-simplices cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B6 and C6 simple
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s. The 60 roots of D6 can be geometrically folded into H3 (
Icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
), as to , creating 2 copies of 30-vertex icosidodecahedra, with the
Golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
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.


Alternate names

* birectified hexacross * birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)


Cartesian coordinates

Cartesian coordinates 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 ...
for the vertices of a rectified hexacross, centered at the origin, edge length \sqrt\ are all permutations of: : (±1,±1,±1,0,0,0)


Images

It can also be projected into 3D-dimensions as → , a
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
envelope.


Related polytopes

These polytopes are a part a family of 63 Uniform 6-polytopes generated from the B6
Coxeter plane In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which hav ...
, including the regular
6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It ...
or
6-orthoplex In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 Vertex (geometry), vertices, 60 Edge (geometry), edges, 160 triangle Face (geometry), faces, 240 tetrahedron Cell (mathematics), cells, 192 5-cell ''4-faces'', and 64 ...
.


Notes


References

* H.S.M. Coxeter: ** 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
wiley.com
*** (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