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Like all polytopes, 4-polytopes may be classified based on properties like "convexity" and "symmetry". A 4-polytope is convex if its boundary (including its cells, faces and edges) does not intersect itself and the line segment joining any two points of the 4-polytope is contained in the 4-polytope or its interior; otherwise, it is non-convex .
The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. [1] He discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell.
The polytopes of rank 2 (2-polytopes) are called polygons.Regular polygons are equilateral and cyclic.A p-gonal regular polygon is represented by Schläfli symbol {p}.. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular.
The most obvious family of prismatic 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytopes are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells).
A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples.
Net. In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C 120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron [1] and hecatonicosahedroid.
Net. In four-dimensional geometry, the 24-cell is the convex regular 4-polytope [1] (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C 24, or the icositetrachoron, [2] octaplex (short for "octahedral complex"), icosatetrahedroid, [3] octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.
Coxeter lists 32 regular compounds of regular 4-polytopes in his book Regular Polytopes. [3] McMullen adds six in his paper New Regular Compounds of 4-Polytopes, in which he also proves that the list is now complete. [4]