Euler Characteristic Calculator

Euler Characteristic (χ): -

Genus (g): -

Surface Classification: -

Understanding Euler Characteristic

What is Euler Characteristic?

The Euler characteristic (χ) is a topological invariant that describes the shape of a topological space regardless of how it is bent or stretched.

χ = V - E + F

For orientable surfaces: χ = 2 - 2g

For non-orientable surfaces: χ = 2 - g

where:

  • V = number of vertices
  • E = number of edges
  • F = number of faces
  • g = genus (number of "holes")

Surface Classification

  • Orientable Surfaces:
    • Sphere: χ = 2, g = 0
    • Torus: χ = 0, g = 1
    • Double Torus: χ = -2, g = 2
    • n-Torus: χ = 2-2n, g = n
  • Non-orientable Surfaces:
    • Projective Plane: χ = 1, g = 1
    • Klein Bottle: χ = 0, g = 2
    • Connected Sum: χ₁ + χ₂ - 2

Applications and Properties

Polyhedra

Regular polyhedra have χ = 2

Cell Complexes

Generalizes to higher dimensions

Poincaré Duality

Relates homology groups

Index Theory

Connects to vector fields