The Mathematics of the Coffee Cup and Donut – Topology

Topology studies properties preserved under continuous deformation (stretching, bending, but not cutting or gluing). A coffee cup with one handle is topologically equivalent to a doughnut (torus) – both have one hole. A sphere is equivalent to a cube, but not to a torus. The branch began with Euler’s polyhedron formula V – E + F = 2 for convex polyhedra. Genus (number of holes) is a topological invariant. Topology is now used to analyse data (persistent homology), study phase transitions in materials, and understand the shape of the universe.