Rectified 6-cube
Encyclopedia
6-cube |
Rectified 6-cube |
Birectified 6-cube |
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Birectified 6-orthoplex |
Rectified 6-orthoplex |
6-orthoplex |
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| Orthogonal projections in A6 Coxeter plane | |||
|---|---|---|---|
In six-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 rectified 6-cube is a convex uniform 6-polytope, being a rectification
Rectification (geometry)
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...
of the regular 6-cube.
There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-ocube are located in the square face centers of the 6-cube.
Rectified 6-cube
| Rectified 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t1{4,3,3,3,3} |
| 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 |
|
| 5-faces | 76 |
| 4-faces | 444 |
| Cells | 1120 |
| Faces | 1520 |
| Edges | 960 |
| Vertices | 192 |
| 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:... |
5-cell prism |
| Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets... |
Dodecagon Dodecagon In geometry, a dodecagon is any polygon with twelve sides and twelve angles.- Regular dodecagon :It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°... |
| 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 |
B6, [3,3,3,3,4] D6, [33,1,1] |
| 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
The rectified 6-cube may be constructed from the 6-cube by truncatingRectification (geometry)
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...
its vertices at the midpoints of its edges.
Coordinates
The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of:
Birectified 6-cube
| Birectified 6-cube | |
|---|---|
| Type | uniform polypeton |
| Schläfli symbol | t2{4,3,3,3,3} |
| 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 |
|
| 5-faces | 76 |
| 4-faces | 636 |
| Cells | 2080 |
| Faces | 3200 |
| Edges | 1920 |
| Vertices | 240 |
| 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:... |
{4}x{3,3} duoprism |
| 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 |
B6, [3,3,3,3,4] D6, [33,1,1] |
| 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
The birectified 6-cube may be constructed from the 6-cube by truncatingRectification (geometry)
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...
its vertices at the midpoints of its edges.
Coordinates
The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of: