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Just as a 3D shape can be projected onto a flat sheet, so a 4-D shape can be projected onto 3-space or even onto a flat sheet. One common projection is a Schlegel diagram which uses stereographic projection of points on the surface of a 3-sphere into three dimensions, connected by straight edges, faces, and cells drawn in 3-space.
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.
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 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]
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]
For 4-polytope: see Polychoron: or Category:4-polytopes: For Polyteron or Polytera: see 5-polytope: or Category:5-polytopes: For Polypeton or Polypeta: see 6-polytope: or Category:6-polytopes: For Polyexon or Polyexa: see 7-polytope: or Category:7-polytopes: For Polyzetton or Polyzetta: see 8-polytope: or Category:8-polytopes: For Polyyotton or ...