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Euler Characteristic
★★★★☆Undergraduate
📖Definition
The Euler characteristic is a topological invariant, calculated for polyhedra as vertices - edges + faces.
📐Formulas
χ = V - E + F
Euler's formula for polyhedra
χ(S²) = 2, χ(T²) = 0
Euler characteristic of sphere and torus
χ = 2 - 2g
Euler characteristic of genus g surface
✏️Examples
예제 1
Calculate the Euler characteristic of a cube.
예제 2
What is the Euler characteristic of a torus?
📜History
Discovered by: Leonhard Euler (1752)
Euler discovered V - E + F = 2 for polyhedra.
⚡Applications
Geometry
Surface classification
Computer Graphics
Mesh validation
Chemistry
Molecular structure analysis
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