6-simplex |
6-orthoplex, 3_{11} |
6-cube (Hexeract) |
2_{21} |
Expanded 6-simplex |
Rectified 6-orthoplex |
6-demicube 1_{31} (Demihexeract) |
1_{22} |
In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.
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:
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]}
6-polytopes may be classified by properties like "convexity" and "symmetry".
Main article: List of regular polytopes § Convex_5 |
Main article: Uniform 6-polytope |
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 three convex regular 6-polytopes, their elements are:
Name | Schläfli symbol |
Coxeter diagram |
Vertices | Edges | Faces | Cells | 4-faces | 5-faces | Symmetry (order) |
---|---|---|---|---|---|---|---|---|---|
6-simplex | {3,3,3,3,3} | 7 | 21 | 35 | 35 | 21 | 7 | A_{6} (720) | |
6-orthoplex | {3,3,3,3,4} | 12 | 60 | 160 | 240 | 192 | 64 | B_{6} (46080) | |
6-cube | {4,3,3,3,3} | 64 | 192 | 240 | 160 | 60 | 12 | B_{6} (46080) |
Main article: Uniform 6-polytope |
Here are six simple uniform convex 6-polytopes, including the 6-orthoplex repeated with its alternate construction.
Name | Schläfli symbol(s) |
Coxeter diagram(s) |
Vertices | Edges | Faces | Cells | 4-faces | 5-faces | Symmetry (order) |
---|---|---|---|---|---|---|---|---|---|
Expanded 6-simplex | t_{0,5}{3,3,3,3,3} | 42 | 210 | 490 | 630 | 434 | 126 | 2×A_{6} (1440) | |
6-orthoplex, 3_{11} (alternate construction) |
{3,3,3,3^{1,1}} | 12 | 60 | 160 | 240 | 192 | 64 | D_{6} (23040) | |
6-demicube | {3,3^{3,1}} h{4,3,3,3,3} |
32 | 240 | 640 | 640 | 252 | 44 | D_{6} (23040) ½B_{6} | |
Rectified 6-orthoplex | t_{1}{3,3,3,3,4} t_{1}{3,3,3,3^{1,1}} |
60 | 480 | 1120 | 1200 | 576 | 76 | B_{6} (46080) 2×D_{6} | |
2_{21} polytope | {3,3,3^{2,1}} | 27 | 216 | 720 | 1080 | 648 | 99 | E_{6} (51840) | |
1_{22} polytope | {3,3^{2,2}} | or |
72 | 720 | 2160 | 2160 | 702 | 54 | 2×E_{6} (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 2_{22} honeycomb,, has 1_{22} polytope is the vertex figure and 2_{21} facets.