Topological Genus & Deformation

Surfaces classified by genus g — Euler characteristic χ = 2 − 2g

χ = V − E + F = 2 − 2g  |  sphere: g=0, χ=2  |  torus: g=1, χ=0
Sphere ≅ Cube ≅ Tetrahedron Torus ≅ Coffee Mug ≅ Donut Klein Bottle: non-orientable
Topology studies properties preserved under continuous deformation (no tearing). The genus g counts the number of "handles" — a sphere has 0, torus has 1, pretzel has 3. The Euler characteristic χ = V−E+F is a topological invariant: same for any triangulation. A coffee mug is homeomorphic to a torus — continuously deformable into it.