Truncated 7-cube

Orthogonal projections in BC₇ Coxeter plane
7-cube	Truncated 7-cube	Bitruncated 7-cube	Tritruncated 7-cube
7-orthoplex	Truncated 7-orthoplex Truncated 7-orthoplex In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated...	Bitruncated 7-orthoplex	Tritruncated 7-orthoplex

In seven-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 7-cube is a convex uniform 7-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 7-cube.

There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square faces of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic

Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex.

Truncated 7-cube
Type	uniform polyexon
Schläfli symbol	t_0,1{4,3,3,3,3,3}
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	3136
Vertices	896
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 5-simplex 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	BC₇, [3,3,3,3,3,4]
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 truncated 7-cube, centered at the origin, are all sign and coordinate 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

(1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)

Bitruncated 7-cube

Bitruncated 7-cube
Type	uniform polyexon
Schläfli symbol	t_1,2{4,3,3,3,3,3}
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	9408
Vertices	2688
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	BC₇, [3,3,3,3,3,4] D₇
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 7-cube, centered at the origin, are all sign and coordinate permutation

Permutation

s of

(±2,±2,±2,±2,±2,±1,0)

Tritruncated 7-cube

Tritruncated 7-cube
Type	uniform polyexon
Schläfli symbol	t_2,3{4,3,3,3,3,3}
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	13440
Vertices	3360
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	BC₇, [3,3,3,3,3,4] D₇
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 tritruncated 7-cube, centered at the origin, are all sign and coordinate permutation

Permutation

s of

(±2,±2,±2,±2,±1,0,0)

External links

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