6.2 Facial structure of polytopes

Facial structure of polytopes is all about understanding the building blocks of these geometric shapes. From vertices to edges to faces, each element plays a crucial role in defining a polytope's form and properties.

Diving deeper, we explore face lattices, which show how different parts of a polytope relate to each other. We also look at special types of polytopes like simplices, cubes, and cross-polytopes, each with their own unique facial structures.

Facial Structure of Polytopes

Faces of convex polytopes

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• Faces of a convex polytope intersect polytope with supporting hyperplane resulting in empty set and polytope itself dimension ranges from -1 (empty set) to d (full polytope) (cube, tetrahedron)

• Proper faces exclude empty set and polytope itself dimension spans 0 (vertices) to d-1 (facets) (edges, triangular faces)

• Face lattice partially orders all faces by inclusion represents combinatorial structure of polytope (Hasse diagram)

Properties of face lattices

• Diamond property ensures exactly two faces H exist between faces F and G when dim(G) = dim(F) + 2 (cube: vertex → two edges → face)

• Möbius function $μ(F, G) = (-1)^{dim(G) - dim(F) - 1}$ for F ⊆ G used in combinatorial formulas and Euler characteristic calculations

• Euler-Poincaré formula $\sum_{F} (-1)^{dim(F)} = 0$ sums over all faces F of polytope (tetrahedron: 1 - 4 + 6 - 4 + 1 = 0)

• Duality of face lattices reverses face lattice of polar polytope (cube and octahedron)

Polytope faces as polytopes

• Face definition intersects polytope with supporting hyperplane
• Polytope properties preserved by intersection:
  • Convexity
  • Boundedness
  • Closure under finite intersections of halfspaces
• Induction argument proves faces are polytopes:
  1. Base case: vertices are 0-dimensional polytopes
  2. Inductive step: show k-dimensional face is polytope

Facial structure analysis

• Simplices form faces from every subset of vertices number of k-faces: $\binom{n+1}{k+1}$ face lattice isomorphic to subset lattice (triangle, tetrahedron)

• Cubes create faces by fixing coordinates 2^n vertices, n2^(n-1) edges, 2n facets face lattice isomorphic to signed subset lattice (3-cube, 4-cube)

• Cross-polytopes convex hull of $\pm e_i$ for standard basis vectors 2n vertices, $2^n - 2$ facets face lattice isomorphic to signed subset lattice with at most one sign per coordinate (octahedron, 16-cell)