5simplex (hexateron) 
5orthoplex, 2_{11} (Pentacross) 
5cube (Penteract) 
Expanded 5simplex 
Rectified 5orthoplex 
5demicube. 1_{21} (Demipenteract) 
In geometry, a fivedimensional polytope (or 5polytope or polyteron) is a polytope in fivedimensional space, bounded by (4polytope) facets, pairs of which share a polyhedral cell.
A 5polytope is a closed fivedimensional figure with vertices, edges, faces, and cells, and 4faces. 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 4face is a 4polytope. Furthermore, the following requirements must be met:
The topology of any given 5polytope 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]}
5polytopes may be classified based on properties like "convexity" and "symmetry".
Regular 5polytopes 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 5polytopes:
For the 3 convex regular 5polytopes and three semiregular 5polytope, their elements are:
Name  Schläfli symbol(s) 
Coxeter diagram(s) 
Vertices  Edges  Faces  Cells  4faces  Symmetry (order) 

5simplex  {3,3,3,3}  6  15  20  15  6  A_{5}, (120)  
5cube  {4,3,3,3}  32  80  80  40  10  BC_{5}, (3820)  
5orthoplex  {3,3,3,4} {3,3,3^{1,1}} 

10  40  80  80  32  BC_{5}, (3840) 2×D_{5} 
For three of the semiregular 5polytope, their elements are:
Name  Schläfli symbol(s) 
Coxeter diagram(s) 
Vertices  Edges  Faces  Cells  4faces  Symmetry (order) 

Expanded 5simplex  t_{0,4}{3,3,3,3}  30  120  210  180  162  2×A_{5}, (240)  
5demicube  {3,3^{2,1}} h{4,3,3,3} 

16  80  160  120  26  D_{5}, (1920) ½BC_{5} 
Rectified 5orthoplex  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 5simplex is the vertex figure of the uniform 5simplex honeycomb, . The 5demicube honeycomb, , vertex figure is a rectified 5orthoplex and facets are the 5orthoplex and 5demicube.
Pyramidal 5polytopes, or 5pyramids, can be generated by a 4polytope base in a 4space hyperplane connected to a point off the hyperplane. The 5simplex is the simplest example with a 4simplex base.