The regular convex 4-polytopes are the four-dimensional analogues of the
Platonic solids in three dimensions and the convex
regular polygons in two dimensions. 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 (face-to-face) in a regular fashion, forming the
surface of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of the earth is a closed, curved 2-dimensional space).
Properties Like their 3-dimensional analogues, the convex regular 4-polytopes can be naturally ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is
rounder than its predecessor, enclosing more content within the same radius.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {
p,q,r} in four dimensions}} The 4-simplex (5-cell) has the smallest content, and the 120-cell has the largest. The following table lists some properties of the six convex regular 4-polytopes. The symmetry groups of these 4-polytopes 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.
John Conway advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD).
Norman Johnson advocated the names n-cell, 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"). The
Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analogue of Euler's polyhedral formula: :N_0 - N_1 + N_2 - N_3 = 0\, where
Nk denotes the number of
k-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.). The topology of any given 4-polytope is defined by its
Betti numbers and
torsion coefficients.
As configurations A regular 4-polytope 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 4-polytope. The non-diagonal 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 4-polytope. The configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.
Visualization The following table shows some 2-dimensional projections of these 4-polytopes. Various other visualizations can be found in the external links below. The
Coxeter-Dynkin diagram graphs are also given below the
Schläfli symbol. ==Regular star (Schläfli–Hess) 4-polytopes==