6 polytope

Definition

A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 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:

Characteristics

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

Regular 6-polytopes can be generated from Coxeter groups represented by the Schläfli symbol {p,q,r,s,t} with t {p,q,r,s} 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 3 convex regular 6-polytopes, their elements are:

Uniform 6-polytopes

Here are six simple uniform convex 6-polytopes, including the 6-orthoplex repeated with its alternate construction.

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.