Mathieu Dutour Sikirić

Definitions

• Polytope: The convex hull of a finite set of points in R^d.
• Flag: A sequence of faces F₀, ..., F_d such that F_i ⊆ F_i+1.
• Regular Polytopes: The symmetry group is transitive on the set of flags.
• Regular-faced d-Polytope: All facets are regular polytopes.
• Semiregular d-Polytope: A regular-faced polytope with equivalent vertices (vertex-transitive).
• Archimedean 4-Polytopes: Facets are regular or Archimedean 3-polytopes.

Classification of Regular Polytopes

Derived from H.S.M. Coxeter, "Regular Polytopes" (1963):

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

Classification of Semiregular Polytopes

Based on work by Gosset (1900) and Blind & Blind (1991):

• All regular polytopes
• 0₂₁ (Hypersimplex)
• 1₂₁ (Half-5-cube)
• 2₂₁, 3₂₁, 4₂₁ (Delaunay/Voronoi polytopes of E6, E7, E8 lattices)
• Snub 24-cell
• Octicosahedric polytope (Medial of 600-cell)

Classification of Regular-Faced Polytopes

• All semiregular polytopes
• The 92 Johnson solids (See their z-structure)
• Pyramids and Bipyramids of simplexes, crosspolytopes, and icosahedrons
• All Special cuts of 600-cell

Archimedean 4-Polytopes

• Wythoffian: 45 polytopes obtained via Wythoff's kaleidoscope construction from irreducible reflection groups (A4, B4, H4, F4).
• Prisms: 17 prisms on Platonic/Archimedean solids and the infinite family of prisms on antiprisms.
• Grand Antiprism: A unique vertex-transitive polytope with 100 vertices, formed by removing two 10-vertex circuits from a 600-cell.