6-polytope

Graphs of three regular and five Uniform 6-polytopes

6-simplex

6-orthoplex, 311

6-cube (Hexeract)

221

Expanded 6-simplex

Rectified 6-orthoplex

6-demicube 131
(Demihexeract)

122

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 (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.[1]

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.[1]

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.[1]

Classification

6-polytopes may be classified by properties like "convexity" and "symmetry".

Main article: Uniform 6-polytope

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:

NameSchläfli
symbol
Coxeter
diagram
VerticesEdgesFacesCells4-faces5-facesSymmetry (order)
6-simplex{3,3,3,3,3}7213535217A6 (720)
6-orthoplex{3,3,3,3,4}126016024019264B6 (46080)
6-cube{4,3,3,3,3}641922401606012B6 (46080)

Uniform 6-polytopes

Main article: Uniform 6-polytope

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

NameSchläfli
symbol(s)
Coxeter
diagram(s)
VerticesEdgesFacesCells4-faces5-facesSymmetry (order)
Expanded 6-simplext0,5{3,3,3,3,3}422104906304341262×A6 (1440)
6-orthoplex, 311
(alternate construction)
{3,3,3,31,1}126016024019264D6 (23040)
6-demicube{3,33,1}
h{4,3,3,3,3}

3224064064025244D6 (23040)
½B6
Rectified 6-orthoplext1{3,3,3,3,4}
t1{3,3,3,31,1}

604801120120057676B6 (46080)
2×D6
221 polytope{3,3,32,1}27216720108064899E6 (51840)
122 polytope{3,32,2}
or
7272021602160702542×E6 (103680)

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.

References

  1. 1.0 1.1 1.2 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.

External links