6.3 Euler's formula and f-vectors

Euler's formula connects vertices, edges, and faces in convex polytopes, revealing intrinsic relationships between face dimensions. This fundamental concept extends from 3D to higher dimensions, serving as a topological invariant across various polytope shapes.

F-vectors and h-vectors provide different ways to represent polytope structure. While f-vectors count faces of each dimension, h-vectors transform this data algebraically. Both vectors offer insights into polytope symmetries and patterns, with practical applications in computation and classification.

Euler's Formula and Polytope Vectors

Euler's formula for polytopes

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• Euler's formula connects vertices, edges, and faces in convex polytopes
• 3D polytopes: $V - E + F = 2$ balances structural elements
• 4D polytopes: $V - E + F - C = 1$ introduces cells (C) for higher dimension
• General form: $f_0 - f_1 + f_2 - ... + (-1)^{d-1}f_{d-1} = 1 - (-1)^d$ extends to d-dimensions
• Proof uses induction starting with tetrahedron (4 vertices, 6 edges, 4 faces)
• Adding vertices maintains formula validity through careful face counting
• Formula serves as topological invariant across convex polytopes (cube, dodecahedron)
• Reveals intrinsic relationships between face dimensions in polytopes

F-vector and h-vector relationship

• F-vector $(f_0, f_1, ..., f_{d-1})$ counts faces of each dimension (cube: 8 vertices, 12 edges, 6 faces)
• H-vector $(h_0, h_1, ..., h_d)$ algebraically transforms f-vector data
• H-vector calculation: $h_k = \sum_{i=0}^k (-1)^{k-i} \binom{d-i}{d-k} f_{i-1}$ combines combinatorics with face counts
• Both vectors contain equivalent information presented differently (cube, octahedron)
• H-vector often reveals hidden symmetries and structural patterns in polytopes
• F-vector recoverable from h-vector through inverse transformation

Computation of polytope vectors

• F-vector computation:
  1. Count vertices $(f_0)$ (cube: 8)
  2. Enumerate edges $(f_1)$ (cube: 12)
  3. Tally faces of each dimension $(f_2, ..., f_{d-1})$ (cube: 6 square faces)
• H-vector derivation:
  1. Determine f-vector
  2. Apply transformation formula for each $h_k$
• Tetrahedron yields $f = (4, 6, 4)$, $h = (1, 1, 1, 1)$ showcasing simplicial structure
• Cube produces $f = (8, 12, 6)$, $h = (1, 3, 3, 1)$ reflecting its symmetry
• Octahedron gives $f = (6, 12, 8)$, $h = (1, 3, 3, 1)$ demonstrating duality with cube

Dehn-Sommerville equations and implications

• Dehn-Sommerville equations constrain f-vectors of simplicial polytopes
• Equation form: $f_{d-i-1} = \sum_{j=0}^i (-1)^{i-j} \binom{d-j}{d-i} f_{j-1}$ for $0 \leq i < \lfloor d/2 \rfloor$
• Reduce independent f-vector entries especially in even dimensions
• Simplify h-vector calculations through linear relationships
• Imply h-vector symmetry: $h_i = h_{d-i}$ for all $i$ (octahedron, icosahedron)
• Aid in polytope classification based on face number patterns
• Prove lower bound theorems for face numbers in specific polytope classes