5-simplex (hexateron) |
5-orthoplex, 2_{11} (Pentacross) |
5-cube (Penteract) |
Expanded 5-simplex |
Rectified 5-orthoplex |
5-demicube. 1_{21} (Demipenteract) |
In geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.
A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five 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, and a 4-face is a 4-polytope. Furthermore, the following requirements must be met:
The topology of any given 5-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, 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]}
5-polytopes may be classified based on properties like "convexity" and "symmetry".
Main article: Uniform 5-polytope |
Main article: List_of_regular_polytopes § Convex_4 |
Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facets around each face.
There are exactly three such convex regular 5-polytopes:
For the 3 convex regular 5-polytopes and three semiregular 5-polytope, their elements are:
Name | Schläfli symbol(s) |
Coxeter diagram(s) |
Vertices | Edges | Faces | Cells | 4-faces | Symmetry (order) |
---|---|---|---|---|---|---|---|---|
5-simplex | {3,3,3,3} | 6 | 15 | 20 | 15 | 6 | A_{5}, (120) | |
5-cube | {4,3,3,3} | 32 | 80 | 80 | 40 | 10 | BC_{5}, (3820) | |
5-orthoplex | {3,3,3,4} {3,3,3^{1,1}} |
10 | 40 | 80 | 80 | 32 | BC_{5}, (3840) 2×D_{5} |
Main article: Uniform 5-polytope |
For three of the semiregular 5-polytope, their elements are:
Name | Schläfli symbol(s) |
Coxeter diagram(s) |
Vertices | Edges | Faces | Cells | 4-faces | Symmetry (order) |
---|---|---|---|---|---|---|---|---|
Expanded 5-simplex | t_{0,4}{3,3,3,3} | 30 | 120 | 210 | 180 | 162 | 2×A_{5}, (240) | |
5-demicube | {3,3^{2,1}} h{4,3,3,3} |
16 | 80 | 160 | 120 | 26 | D_{5}, (1920) ½BC_{5} | |
Rectified 5-orthoplex | t_{1}{3,3,3,4} t_{1}{3,3,3^{1,1}} |
40 | 240 | 400 | 240 | 42 | BC_{5}, (3840) 2×D_{5} |
The expanded 5-simplex is the vertex figure of the uniform 5-simplex honeycomb, . The 5-demicube honeycomb, , vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube.
Pyramidal 5-polytopes, or 5-pyramids, can be generated by a 4-polytope base in a 4-space hyperplane connected to a point off the hyperplane. The 5-simplex is the simplest example with a 4-simplex base.