• A polytope is the convex hull of a finite set of points in the d-dimensional space R^d.
• a flag is a sequence F0, ..., Fd of faces of P with Fi included in Fi+1.
• Regular polytopes: the symmetry group of P is transitive on the set of flags of P.
• Regular-faced d-polytope: the facets are regular polytopes.
• Semi-regular d-polytope: regular-faced d-polytope with equivalent vertices (i.e. the group of symmetries of the polytope is transitive on vertices)
• Archimedean 4-polytopes: facets are regular or archimedean 3-polytopes.

• The n-simplex
• The n-cube
• The n-cross-polytope
• The 24-cell
• The 120-cell
• The 600-cell

H.S.M. Coxeter, "Regular polytopes" New York 1963.

• All regular polytopes
• 0_21 also called hypersimplex
• 1_21, half-5-cube
• 2_21, Delaunay polytope of the root lattice E6
• 3_21, Delaunay polytope of the root lattice E7
• 4_21, Voronoi polytope of the root lattice E8
• snub 24-cell
• octicosahedric polytope, i.e. the medial of 600-cell.

T.Gosset, "On the regular and semiregular figures in spaces of $n$ dimensions", Messenger of Mathematics 29 (1900) 43--48.

• All semi-regular polytopes
• 92 Johnson solids Their z-structure
• Pyr(n-crosspolytope)
• BPyr(n-simplex)
• Pyr(Ico)
• BPyr(Ico)
• 0_21+Pyr(3-crosspolytope)
• All special cuts of 600-cell Their z-structure

• 45 polytopes obtained by Wythoff's kaleidoscope construction from 4-dimensional irreducible reflection (point) groups;
• There are 4 irreducible Coxeter reflexion groups in dimension 4: A4 (simplex), B4 (hypercube and hyperoctahedron), H4 (600-cell and 120-cell), F4 (24-cell).
• See Isometric embeddings of Archimedean Wythoff polytopes into hypercubes and half-cubes for theoretical detail of the construction and Zigzag structure of complexes for the details of the situation in dimension 4 and how the 45 polytopes are obtained.
• 17 prisms on Platonic (other than Cube) and Archimedean solids;
• prisms on the Antiprism;
• a doubly infinite set of polytopes, which are direct products of two regular polygons (if one of polygons is a square, then we get prisms on 3-dimensional prisms);
• the snub 24-cell;(explained also in Regular polytopes )
• a new polytope, called Grand Antiprism See below a short desciption of how to construct it
• The 600-cell is a regular polytope having 600 simplicial facets belonging to 120 vertices.
• The set of vertex adjacent to a given vertex form an isosahedron. So, given a vertex v and another vertex w adjacent to v, there exist an unique opposite vertex w' of w relatively to v
• There exist a circuit v1, ..., v10 of vertices with v_{i+1} being opposite to v_{i-1} relatively to v_{i}
• Given one such circuit, the antipodal make another such circuit.
• Take one circuit and its antipodal and remove it from the list of vertices of 600-cell.
• The obtained polytope has 100 vertices, its facet set consist of 300 simplices and 20 5-antiprisms. This polytope is vertex transitive.