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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 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.
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 similarly self-intersecting polytope in any number of dimensions is called a star polytope. A regular polytope {p,q,r,...,s,t} is a star polytope if either its facet {p,q,...s} or its vertex figure {q,r,...,s,t} is a star polytope. In four dimensions, the 10 regular star polychora are called the Schläfli–Hess polychora.
In four dimensions the regular 4-polytopes include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star Schläfli-Hess 4-polytopes, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.
Regular star 4-polytopes (star polyhedron cells and/or vertex figures) 1852 : Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures { 5 / 2 ,5} and {5, 5 / 2 } .
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.