In mathematics, a regular 4polytope or regular polychoron is a regular fourdimensional polytope. They are the fourdimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star regular 4polytopes, giving a total of sixteen.
The convex regular 4polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid19th century.^{[1]} He discovered that there are precisely six such figures.
Schläfli also found four of the regular star 4polytopes: the grand 120cell, great stellated 120cell, grand 600cell, and great grand stellated 120cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zerohole tori: F − E + V = 2). That excludes cells and vertex figures such as the great dodecahedron {5,5/2} and small stellated dodecahedron {5/2,5}.
Edmund Hess (1843–1903) published the complete list in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder.
The existence of a regular 4polytope is constrained by the existence of the regular polyhedra which form its cells and a dihedral angle constraint
to ensure that the cells meet to form a closed 3surface.
The six convex and ten star polytopes described are the only solutions to these constraints.
There are four nonconvex Schläfli symbols {p,q,r} that have valid cells {p,q} and vertex figures {q,r}, and pass the dihedral test, but fail to produce finite figures: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}.
The regular convex 4polytopes are the fourdimensional analogues of the Platonic solids in three dimensions and the convex regular polygons in two dimensions.
Each convex regular 4polytope is bounded by a set of 3dimensional cells which are all Platonic solids of the same type and size. These are fitted together along their respective faces (facetoface) in a regular fashion, forming the surface of the 4polytope which is a closed, curved 3dimensional space (analogous to the way the surface of the earth is a closed, curved 2dimensional space).
Like their 3dimensional analogues, the convex regular 4polytopes can be naturally 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 within the same radius.^{[2]} The 4simplex (5cell) has the smallest content, and the 120cell has the largest.
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  
Mirror dihedrals  𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2  𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2  𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  
Graph  
Vertices  5 tetrahedral  8 octahedral  16 tetrahedral  24 cubical  120 icosahedral  600 tetrahedral  
Edges  10 triangular  24 square  32 triangular  96 triangular  720 pentagonal  1200 triangular  
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  675 in 120cell  2 16cells  3 8cells  25 24cells  10 600cells  
Great polygons  2 squares x 3  4 rectangles x 4  4 hexagons x 4  12 decagons x 6  100 irregular hexagons x 4  
Petrie polygons  1 pentagon x 2  1 octagon x 3  2 octagons x 4  2 dodecagons x 4  4 30gons x 6  20 30gons x 4  
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
The following table lists some properties of the six convex regular 4polytopes. The symmetry groups of these 4polytopes are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group.
Names  Image  Family  Schläfli Coxeter 
V  E  F  C  Vert. fig. 
Dual  Symmetry group  

5cell pentachoron pentatope 4simplex 
nsimplex (A_{n} family) 
{3,3,3} 
5  10  10 {3} 
5 {3,3} 
{3,3}  selfdual  A_{4} [3,3,3] 
120  
16cell hexadecachoron 4orthoplex 
northoplex (B_{n} family) 
{3,3,4} 
8  24  32 {3} 
16 {3,3} 
{3,4}  8cell  B_{4} [4,3,3] 
384  
8cell octachoron tesseract 4cube 
hypercube ncube (B_{n} family) 
{4,3,3} 
16  32  24 {4} 
8 {4,3} 
{3,3}  16cell  
24cell icositetrachoron octaplex polyoctahedron (pO) 
F_{n} family  {3,4,3} 
24  96  96 {3} 
24 {3,4} 
{4,3}  selfdual  F_{4} [3,4,3] 
1152  
600cell hexacosichoron tetraplex polytetrahedron (pT) 
npentagonal polytope (H_{n} family) 
{3,3,5} 
120  720  1200 {3} 
600 {3,3} 
{3,5}  120cell  H_{4} [5,3,3] 
14400  
120cell hecatonicosachoron dodecacontachoron dodecaplex polydodecahedron (pD) 
npentagonal polytope (H_{n} family) 
{5,3,3} 
600  1200  720 {5} 
120 {5,3} 
{3,3}  600cell 
John Conway advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD).^{[3]}
Norman Johnson advocated the names ncell, or pentachoron, hexadecachoron, tesseract or octachoron, icositetrachoron, hexacosichoron, and hecatonicosachoron (or dodecacontachoron), coining the term polychoron being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the Greek roots poly ("many") and choros ("room" or "space").^{[4]}^{[5]}
The Euler characteristic for all 4polytopes is zero, we have the 4dimensional analogue of Euler's polyhedral formula:
where N_{k} denotes the number of kfaces in the polytope (a vertex is a 0face, an edge is a 1face, etc.).
The topology of any given 4polytope is defined by its Betti numbers and torsion coefficients.^{[6]}
A regular 4polytope can be completely described as a configuration matrix containing counts of its component elements. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers (upper left to lower right) say how many of each element occur in the whole 4polytope. The nondiagonal numbers say how many of the column's element occur in or at the row's element. For example, there are 2 vertices in each edge (each edge has 2 vertices), and 2 cells meet at each face (each face belongs to 2 cells), in any regular 4polytope. The configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.^{[7]}^{[8]}
5cell {3,3,3} 
16cell {3,3,4} 
8cell {4,3,3} 
24cell {3,4,3} 
600cell {3,3,5} 
120cell {5,3,3} 

The following table shows some 2dimensional projections of these 4polytopes. Various other visualizations can be found in the external links below. The CoxeterDynkin diagram graphs are also given below the Schläfli symbol.
A_{4} = [3,3,3]  B_{4} = [4,3,3]  F_{4} = [3,4,3]  H_{4} = [5,3,3]  

5cell  16cell  8cell  24cell  600cell  120cell 
{3,3,3}  {3,3,4}  {4,3,3}  {3,4,3}  {3,3,5}  {5,3,3} 
Solid 3D orthographic projections  
Tetrahedral envelope (cell/vertexcentered) 
Cubic envelope (cellcentered) 
Cubic envelope (cellcentered) 
Cuboctahedral envelope (cellcentered) 
Pentakis icosidodecahedral envelope (vertexcentered) 
Truncated rhombic triacontahedron envelope (cellcentered) 
Wireframe Schlegel diagrams (Perspective projection)  
Cellcentered 
Cellcentered 
Cellcentered 
Cellcentered 
Vertexcentered 
Cellcentered 
Wireframe stereographic projections (3sphere)  
The Schläfli–Hess 4polytopes are the complete set of 10 regular selfintersecting star polychora (fourdimensional polytopes).^{[10]} They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol {p,q,r} in which one of the numbers is 5/2. They are thus analogous to the regular nonconvex Kepler–Poinsot polyhedra, which are in turn analogous to the pentagram.
Their names given here were given by John Conway, extending Cayley's names for the Kepler–Poinsot polyhedra: along with stellated and great, he adds a grand modifier. Conway offered these operational definitions:
John Conway names the 10 forms from 3 regular celled 4polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral 600cell), pI=polyicosahedron {3,5,5/2} (an icosahedral 120cell), and pD=polydodecahedron {5,3,3} (a dodecahedral 120cell), with prefix modifiers: g, a, and s for great, (ag)grand, and stellated. The final stellation, the great grand stellated polydodecahedron contains them all as gaspD.
All ten polychora have [3,3,5] (H_{4}) hexacosichoric symmetry. They are generated from 6 related Goursat tetrahedra rationalorder symmetry groups: [3,5,5/2], [5,5/2,5], [5,3,5/2], [5/2,5,5/2], [5,5/2,3], and [3,3,5/2].
Each group has 2 regular starpolychora, except for two groups which are selfdual, having only one. So there are 4 dualpairs and 2 selfdual forms among the ten regular star polychora.
Note:
The cells (polyhedra), their faces (polygons), the polygonal edge figures and polyhedral vertex figures are identified by their Schläfli symbols.
Name Conway (abbrev.) 
Orthogonal projection 
Schläfli Coxeter 
C {p, q} 
F {p} 
E {r} 
V {q, r} 
Dens.  χ 

Icosahedral 120cell polyicosahedron (pI) 
{3,5,5/2} 
120 {3,5} 
1200 {3} 
720 {5/2} 
120 {5,5/2} 
4  480  
Small stellated 120cell stellated polydodecahedron (spD) 
{5/2,5,3} 
120 {5/2,5} 
720 {5/2} 
1200 {3} 
120 {5,3} 
4  −480  
Great 120cell great polydodecahedron (gpD) 
{5,5/2,5} 
120 {5,5/2} 
720 {5} 
720 {5} 
120 {5/2,5} 
6  0  
Grand 120cell grand polydodecahedron (apD) 
{5,3,5/2} 
120 {5,3} 
720 {5} 
720 {5/2} 
120 {3,5/2} 
20  0  
Great stellated 120cell great stellated polydodecahedron (gspD) 
{5/2,3,5} 
120 {5/2,3} 
720 {5/2} 
720 {5} 
120 {3,5} 
20  0  
Grand stellated 120cell grand stellated polydodecahedron (aspD) 
{5/2,5,5/2} 
120 {5/2,5} 
720 {5/2} 
720 {5/2} 
120 {5,5/2} 
66  0  
Great grand 120cell great grand polydodecahedron (gapD) 
{5,5/2,3} 
120 {5,5/2} 
720 {5} 
1200 {3} 
120 {5/2,3} 
76  −480  
Great icosahedral 120cell great polyicosahedron (gpI) 
{3,5/2,5} 
120 {3,5/2} 
1200 {3} 
720 {5} 
120 {5/2,5} 
76  480  
Grand 600cell grand polytetrahedron (apT) 
{3,3,5/2} 
600 {3,3} 
1200 {3} 
720 {5/2} 
120 {3,5/2} 
191  0  
Great grand stellated 120cell great grand stellated polydodecahedron (gaspD) 
{5/2,3,3} 
120 {5/2,3} 
720 {5/2} 
1200 {3} 
600 {3,3} 
191  0 