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Non-convex uniform 4-polytopes (10 + unknown) The great grand stellated 120-cell is the largest of 10 regular star 4-polytopes, having 600 vertices. 10 (regular) Schläfli-Hess polytopes 57 hyperprisms built on nonconvex uniform polyhedra
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 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.
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]
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.
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).
The 120-cell is the compound of all five of the other regular convex 4-polytopes. [20] All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell. [b] It is a four-dimensional jigsaw puzzle in which all those polytopes are the parts. [21]
This category contains polytopes of 4-space, and honeycombs of 3-space. Subcategories. This category has the following 2 subcategories, out of 2 total. 0–9.