Convex regular 4polytope facts for kids
In mathematics, a convex regular 4polytope (or polychoron) is 4dimensional (4D) polytope which is both regular and convex. These are the fourdimensional analogs of the Platonic solids (in three dimensions) and the regular polygons (in two dimensions).
These polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid19th century. Schläfli discovered that there are precisely six such figures. Five of these may be thought of as higher dimensional analogs of the Platonic solids. There is one additional figure (the 24cell) which has no threedimensional equivalent.
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 in a regular fashion.
Properties
The following tables lists some properties of the six convex regular polychora. The symmetry groups of these polychora 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  Family  Schläfli symbol 
Vertices  Edges  Faces  Cells  Vertex figures  Dual polytope  Symmetry group  

Pentachoron 5cell pentatope hyperpyramid hypertetrahedron 4simplex 
simplex (nsimplex) 
{3,3,3}  5  10  10 triangles 
5 tetrahedra 
tetrahedra  (selfdual)  A_{4}  120 
Tesseract octachoron 8cell hypercube 4cube 
hypercube (ncube) 
{4,3,3}  16  32  24 squares 
8 cubes 
tetrahedra  16cell  B_{4}  384 
Hexadecachoron 16cell orthoplex hyperoctahedron 4orthoplex 
crosspolytope (northoplex) 
{3,3,4}  8  24  32 triangles 
16 tetrahedra 
octahedra  tesseract  B_{4}  384 
Icositetrachoron 24cell octaplex polyoctahedron 
{3,4,3}  24  96  96 triangles 
24 octahedra 
cubes  (selfdual)  F_{4}  1152  
Hecatonicosachoron 120cell dodecaplex hyperdodecahedron polydodecahedron 
{5,3,3}  600  1200  720 pentagons 
120 dodecahedra 
tetrahedra  600cell  H_{4}  14400  
Hexacosichoron 600cell tetraplex hypericosahedron polytetrahedron 
{3,3,5}  120  720  1200 triangles 
600 tetrahedra 
icosahedra  120cell  H_{4}  14400 
Since the boundaries of each of these figures is topologically equivalent to a 3sphere, whose Euler characteristic is zero, we have the 4dimensional analog 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.).
Visualizations
The following table shows some 2 dimensional projections of these polytopes. Various other visualizations can be found in the other websites below. The CoxeterDynkin diagram graphs are also given below the Schläfli symbol.
5cell  8cell  16cell  24cell  120cell  600cell 

{3,3,3}  {4,3,3}  {3,3,4}  {3,4,3}  {5,3,3}  {3,3,5} 
Wireframe orthographic projections inside Petrie polygons.  
Solid orthographic projections  
tetrahedral envelope (cell/vertexcentered) 
cubic envelope (cellcentered) 
octahedral envelope (vertex centered) 
cuboctahedral envelope (cellcentered) 
truncated rhombic triacontahedron envelope (cellcentered) 
Pentakis icosidodecahedral envelope (vertexcentered) 
Wireframe Schlegel diagrams (Perspective projection)  
(Cellcentered) 
(Cellcentered) 
(Cellcentered) 
(Cellcentered) 
(Cellcentered) 
(Vertexcentered) 
Wireframe stereographic projections (Hyperspherical)  
Related pages
 H. S. M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons Inc., 1969. ISBN: 0471504580.
 H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN: 0486614808.
 D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
 Eric W. Weisstein, Regular polychoron at MathWorld.
Images for kids

This shows the relationships among the fourdimensional starry polytopes. The 2 convex forms and 10 starry forms can be seen in 3D as the vertices of a cuboctahedron.