6-polytope


In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

Definition

A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells, 4-faces, and 5-faces. A vertex is a point where six or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron. A 4-face is a polychoron, and a 5-face is a 5-polytope. Furthermore, the following requirements must be met:
The topology of any given 6-polytope is defined by its Betti numbers and torsion coefficients.
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 6-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.

Classification

6-polytopes may be classified by properties like "convexity" and "symmetry".
Regular 6-polytopes can be generated from Coxeter groups represented by the Schläfli symbol with t 5-polytope facets around each cell.
There are only three such convex regular 6-polytopes:
There are no nonconvex regular polytopes of 5 or more dimensions.
For the three convex regular 6-polytopes, their elements are:
NameSchläfli
symbol
Coxeter
diagram
VerticesEdgesFacesCells4-faces5-facesSymmetry
6-simplex7213535217A6
6-orthoplex126016024019264B6
6-cube641922401606012B6

Uniform 6-polytopes

Here are six simple uniform convex 6-polytopes, including the 6-orthoplex repeated with its alternate construction.
NameSchläfli
symbol
Coxeter
diagram
VerticesEdgesFacesCells4-faces5-facesSymmetry
Expanded 6-simplext0,5422104906304341262×A6
6-orthoplex, 311
126016024019264D6
6-demicube
h

3224064064025244D6
½B6
Rectified 6-orthoplext1
t1

604801120120057676B6
2×D6
221 polytope27216720108064899E6
122 polytope
or
7272021602160702542×E6

The expanded 6-simplex is the vertex figure of the uniform 6-simplex honeycomb,. The 6-demicube honeycomb,, vertex figure is a rectified 6-orthoplex and facets are the 6-orthoplex and 6-demicube. The uniform 222 honeycomb,, has 122 polytope is the vertex figure and 221 facets.