In eight-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 8-orthoplex is a convex

uniform 8-polytope In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets. A uniform 8-polytope is one which is vertex-transitive ...

, 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

8-orthoplex In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 Vertex (geometry), vertices, 112 Edge (geometry), edges, 448 triangle Face (geometry), faces, 1120 tetrahedron Cell (mathematics), cells, 1792 5-cell ''4-faces'', 179 ...

. There are unique 8 degrees of rectifications, the zeroth being the

, and the 7th and last being the

8-cube In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces. It is represented b ...

. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the

tetrahedral In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

cell centers of the 8-orthoplex.

Rectified 8-orthoplex

The rectified 8-orthoplex has 112 vertices. These 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 ...

D₈. 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 28 vertices

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

s cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B₈ and C₈ 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 ...

Related polytopes

The ''rectified 8-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 demiocteractic honeycomb. : or

Alternate names

* rectified octacross * rectified diacosipentacontahexazetton (Acronym: rek) (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 8-orthoplex'', one with the C₈ or ,3⁶Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D₈ or ^5,1,1Coxeter 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 8-orthoplex, centered at the origin, edge length

\sqrt

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

Images

Birectified 8-orthoplex

Alternate names

* birectified octacross * birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)

Cartesian coordinates

for the vertices of a birectified 8-orthoplex, centered at the origin, edge length

\sqrt

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

Images

Trirectified 8-orthoplex

The trirectified 8-orthoplex can

tessellate A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...

space in the quadrirectified 8-cubic honeycomb.

Alternate names

* trirectified octacross * trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)Klitzing, (o3o3o3x3o3o3o4o - tark)

Cartesian coordinates

for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length

\sqrt

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

Images

Notes

References

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

: ** 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. * o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark

External links

Polytopes of Various Dimensions

{{Polytopes 8-polytopes