In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star regular 4-polytopes, giving a total of sixteen.
In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles. The tesseract is one of the six convex regular 4-polytopes.
The tesseract is also called an 8-cell, C8, (regular) octachoron, or cubic prism. It is the four-dimensional measure polytope, taken as a unit for hypervolume. Coxeter labels it the γ4 polytope. The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope.
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}.It is also known as the C600, hexacosichoron, hexacosihedroid and hypericosahedron.It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.
The 600-cell's boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex.Together they form 1200 triangular faces, 720 edges, and 120 vertices.It is the 4-dimensional analogue of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex.Its dual polytope is the 120-cell.