Select Shape
V vertices
4
E edges
6
F faces
4
χ = V − E + F
4 − 6 + 4 = 2
Euler's theorem (1758): For any convex polyhedron homeomorphic to a sphere: V − E + F = 2.
Generalization: χ depends only on topology, not geometry. Deform, subdivide, triangulate — χ is unchanged.
Genus g surface: χ = 2 − 2g. Sphere g=0 → χ=2; Torus g=1 → χ=0; Double-torus g=2 → χ=−2.
Non-orientable: Klein bottle χ=0; projective plane χ=1.
Key insight: Every subdivision (barycentric, edge-flip, stellar) leaves χ invariant — it truly measures the "hole structure" of the space.
Generalization: χ depends only on topology, not geometry. Deform, subdivide, triangulate — χ is unchanged.
Genus g surface: χ = 2 − 2g. Sphere g=0 → χ=2; Torus g=1 → χ=0; Double-torus g=2 → χ=−2.
Non-orientable: Klein bottle χ=0; projective plane χ=1.
Key insight: Every subdivision (barycentric, edge-flip, stellar) leaves χ invariant — it truly measures the "hole structure" of the space.