 5-simplex (hexateron) |  5-orthoplex, 211 (Pentacross) |  5-cube (Penteract) |
 Expanded 5-simplex |  Rectified 5-orthoplex |  5-demicube. 121 (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".
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 | A5, (120) |
| 5-cube | {4,3,3,3} |  | 32 | 80 | 80 | 40 | 10 | BC5, (3820) |
| 5-orthoplex | {3,3,3,4} {3,3,31,1} |   | 10 | 40 | 80 | 80 | 32 | BC5, (3840) 2×D5 |
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 | t0,4{3,3,3,3} | | 30 | 120 | 210 | 180 | 162 | 2×A5, (240) |
| 5-demicube | {3,32,1} h{4,3,3,3} |    | 16 | 80 | 160 | 120 | 26 | D5, (1920) ½BC5 |
| Rectified 5-orthoplex | t1{3,3,3,4} t1{3,3,31,1} | | 40 | 240 | 400 | 240 | 42 | BC5, (3840) 2×D5 |
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.
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n.
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell.
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.
In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence.
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.
In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.
In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).
In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.
In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.