Truncated 8-orthoplex

Orthogonal projections in BC₈ Coxeter plane
8-orthoplex	Truncated 8-orthoplex	Bitruncated 8-orthoplex	Tritruncated 8-orthoplex	Quadritruncated 8-cube
8-cube	Truncated 8-cube Truncated 8-cube In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube...	Bitruncated 8-cube	Tritruncated 8-cube	Quadritruncated 8-cube

In eight-dimensional geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation

Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

of the regular 8-orthoplex.

There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube.

Truncated 8-orthoplex
Type	uniform polyzetton
Schläfli symbol	t_0,1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s
6-faces
5-faces
4-faces
Cells
Faces
Edges	1456
Vertices	224
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	Elongated 6-orthoplex pyramid
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... s	BC₈, [3,3,3,3,3,3,4] D₈, [3^5,1,1]
Properties	convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

Construction

There are two Coxeter group

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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

s associated with the truncated 8-orthoplex, one with the C₈ or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the D₈ or [3^5,1,1] Coxeter group.

Coordinates

Cartesian coordinates for the vertices of a truncated 8-orthoplex, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutation

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

s of

(±2,±1,0,0,0,0,0,0)

Bitruncated 8-orthoplex

Bitruncated 8-orthoplex
Type	uniform polyzetton
Schläfli symbol	t_1,2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... s	BC₈, [3,3,3,3,3,3,4] D₈, [3^5,1,1]
Properties	convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

Coordinates

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutation

Permutation

s of

(±2,±2,±1,0,0,0,0,0)

Tritruncated 8-orthoplex

Tritruncated 8-orthoplex
Type	uniform polyzetton
Schläfli symbol	t_2,3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... s	BC₈, [3,3,3,3,3,3,4] D₈, [3^5,1,1]
Properties	convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

Coordinates

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutation

Permutation

s of

(±2,±2,±2,±1,0,0,0,0)

External links

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