Convex regular 4-polytope

In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional (4D) polytope which is both regular and convex. These are the four-dimensional 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 mid-19th 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 24-cell) which has no three-dimensional equivalent.

Each convex regular 4-polytope is bounded by a set of 3-dimensional 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  Schläflisymbol  Vertices  Edges  Faces  Cells  Vertex figures  Dual polytope Symmetry group Pentachoron5-cellpentatopehyperpyramidhypertetrahedron4-simplex  simplex(n-simplex)  {3,3,3}  5  10  10triangles  5tetrahedra  tetrahedra  (self-dual)  A4  120 Tesseractoctachoron8-cellhypercube4-cube  hypercube(n-cube)  {4,3,3}  16  32  24squares  8cubes  tetrahedra  16-cell  B4  384 Hexadecachoron16-cellorthoplexhyperoctahedron4-orthoplex  cross-polytope(n-orthoplex)  {3,3,4}  8  24  32triangles  16tetrahedra  octahedra  tesseract  B4  384 Icositetrachoron24-celloctaplexpolyoctahedron   {3,4,3}  24  96  96triangles  24octahedra  cubes  (self-dual)  F4  1152 Hecatonicosachoron120-celldodecaplexhyperdodecahedronpolydodecahedron   {5,3,3}  600  1200  720pentagons  120dodecahedra  tetrahedra  600-cell  H4  14400 Hexacosichoron600-celltetraplexhypericosahedronpolytetrahedron   {3,3,5}  120  720  1200triangles  600tetrahedra  icosahedra  120-cell  H4  14400

Since the boundaries of each of these figures is topologically equivalent to a 3-sphere, whose Euler characteristic is zero, we have the 4-dimensional analog of Euler's polyhedral formula:

where Nk denotes the number of k-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, 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 Coxeter-Dynkin diagram graphs are also given below the Schläfli symbol.

 5-cell  8-cell  16-cell  24-cell  120-cell  600-cell {3,3,3}  {4,3,3}  {3,3,4}  {3,4,3}  {5,3,3}  {3,3,5} File:CDel node_1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node_1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node_1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node_1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node_1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node_1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngWireframe orthographic projections inside Petrie polygons. 105px 105px 105px 105px 105px 105pxSolid orthographic projections 105pxtetrahedralenvelope(cell/vertex-centered) 105pxcubic envelope(cell-centered) 105pxoctahedralenvelope(vertex centered) 105pxcuboctahedralenvelope(cell-centered) 105pxtruncated rhombictriacontahedronenvelope(cell-centered) 105pxPentakis icosidodecahedralenvelope(vertex-centered)Wireframe Schlegel diagrams (Perspective projection) 105px(Cell-centered) 105px(Cell-centered) 105px(Cell-centered) 105px(Cell-centered) 105px(Cell-centered) 105px(Vertex-centered)Wireframe stereographic projections (Hyperspherical) 105px 105px 105px 105px 105px 105px

Related pages
Regular polytope
Platonic solid

References

H. S. M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons Inc., 1969. .
H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. .
 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

Other websites

Jonathan Bowers, 16 regular polychora 
Regular 4D Polytope Foldouts 
Catalog of Polytope Images A collection of stereographic projections of 4-polytopes.
A Catalog of Uniform Polytopes
Dimensions 2 hour film about the fourth dimension (contains stereographic projections of all regular polychrons)

Category:Shapes