{3,3,3}  {3,3,4}  {4,3,3} 

5cell Pentatope 4simplex 
16cell Orthoplex 4orthoplex 
8cell Tesseract 4cube 
{3,4,3}  {5,3,3}  {3,3,5} 
24cell Octaplex 
600cell Tetraplex 
120cell Dodecaplex 
In geometry, a 4polytope (sometimes also called a polychoron,^{[1]} polycell, or polyhedroid) is a fourdimensional polytope.^{[2]}^{[3]} It is a connected and closed figure, composed of lowerdimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.^{[4]}
The twodimensional analogue of a 4polytope is a polygon, and the threedimensional analogue is a polyhedron.
Topologically 4polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4polytopes can be cut and unfolded as nets in 3space.
A 4polytope is a closed fourdimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the threedimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in which each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4polytope cannot be subdivided into two or more sets which are also 4polytopes, i.e. it is not a compound.
The convex regular 4polytopes are the fourdimensional analogues of the Platonic solids. The most familiar 4polytope is the tesseract or hypercube, the 4D analogue of the cube.
The convex regular 4polytopes can be ordered by size as a measure of 4dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content^{[5]} within the same radius. The 4simplex (5cell) is the limit smallest case, and the 120cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering.
Regular convex 4polytopes  

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
Schläfli symbol  {3, 3, 3}  {3, 3, 4}  {4, 3, 3}  {3, 4, 3}  {3, 3, 5}  {5, 3, 3}  
Coxeter mirrors  
Graph  
Vertices  5  8  16  24  120  600  
Edges  10  24  32  96  720  1200  
Faces  10 triangles  32 triangles  24 squares  96 triangles  1200 triangles  720 pentagons  
Cells  5 tetrahedra  16 tetrahedra  8 cubes  24 octahedra  600 tetrahedra  120 dodecahedra  
Tori  1 5tetrahedron  2 8tetrahedron  2 4cube  4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  1 16cell  2 16cells  3 8cells  5 24cells x 5  5 600cells x 2  
Great polygons  2 𝝅/2 squares x 3  4 𝝅/2 rectangles x 3  4 𝝅/3 hexagons x 4  12 𝝅/5 decagons x 6  50 𝝅/15 dodecagons x 4  
Petrie polygons  1 pentagon  1 octagon  2 octagons  2 dodecagons  4 30gons  20 30gons  
Isocline polygons  1 {8/2}=2{4} x {8/2}=2{4}  2 {8/2}=2{4} x {8/2}=2{4}  2 {12/2}=2{6} x {12/6}=6{2}  4 {30/2}=2{15} x 30{0}  20 {30/2}=2{15} x 30{0}  
Long radius  1  1  1  1  1  1  
Edge length  √5/√2 ≈ 1.581  √2 ≈ 1.414  1  1  1/ϕ ≈ 0.618  1/√2ϕ^{2} ≈ 0.270  
Short radius  1/4  1/2  1/2  √2/2 ≈ 0.707  1  (√2/2√3φ)^{2} ≈ 0.936  1  (1/2√3φ)^{2} ≈ 0.968  
Area  10•√8/3 ≈ 9.428  32•√3/4 ≈ 13.856  24  96•√3/4 ≈ 41.569  1200•√3/8φ^{2} ≈ 99.238  720•25+10√5/8φ^{4} ≈ 621.9  
Volume  5•5√5/24 ≈ 2.329  16•1/3 ≈ 5.333  8  24•√2/3 ≈ 11.314  600•1/3√8φ^{3} ≈ 16.693  120•2 + φ/2√8φ^{3} ≈ 18.118  
4Content  √5/24•(√5/2)^{4} ≈ 0.146  2/3 ≈ 0.667  1  2  Short∙Vol/4 ≈ 3.907  Short∙Vol/4 ≈ 4.385 
Sectioning  Net  

Projections  
Schlegel  2D orthogonal  3D orthogonal 
4polytopes cannot be seen in threedimensional space due to their extra dimension. Several techniques are used to help visualise them.
Orthogonal projections can be used to show various symmetry orientations of a 4polytope. They can be drawn in 2D as vertexedge graphs, and can be shown in 3D with solid faces as visible projective envelopes.
Just as a 3D shape can be projected onto a flat sheet, so a 4D shape can be projected onto 3space or even onto a flat sheet. One common projection is a Schlegel diagram which uses stereographic projection of points on the surface of a 3sphere into three dimensions, connected by straight edges, faces, and cells drawn in 3space.
Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4polytope reveals a cut "hypersurface" in three dimensions. A sequence of such sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce a smooth animation of these cross sections.
A net of a 4polytope is composed of polyhedral cells that are connected by their faces and all occupy the same threedimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane.
The topology of any given 4polytope is defined by its Betti numbers and torsion coefficients.^{[6]}
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4polytopes, 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.^{[6]}
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4polytopes, and this led to the use of torsion coefficients.^{[6]}
Like all polytopes, 4polytopes may be classified based on properties like "convexity" and "symmetry".
The following lists the various categories of 4polytopes classified according to the criteria above:
Uniform 4polytope (vertextransitive):
Other convex 4polytopes:
Infinite uniform 4polytopes of Euclidean 3space (uniform tessellations of convex uniform cells)
Infinite uniform 4polytopes of hyperbolic 3space (uniform tessellations of convex uniform cells)
Dual uniform 4polytope (celltransitive):
Others:
These categories include only the 4polytopes that exhibit a high degree of symmetry. Many other 4polytopes are possible, but they have not been studied as extensively as the ones included in these categories.
Wikimedia Commons has media related to 4polytopes. 