Topology
What survives stretching and bending, but not cutting
What it is
Topology studies properties that survive continuous deformation - stretching, bending, twisting - but not cutting or gluing. What matters is connectivity and structure, not exact distance or angle. A Möbius strip, torus, and Klein bottle are the classic hands-on examples: surfaces whose "strangeness" (one-sidedness, holes, self-intersection when forced into 3D) comes purely from how they're connected, not from any particular measurement.
History
The field's founding moment is usually traced to Leonhard Euler's 1736 solution to the Seven Bridges of Königsberg problem and his polyhedron formula. August Möbius and Johann Benedict Listing independently described the one-sided strip now named after Möbius in 1858. Felix Klein described his self-intersecting "bottle" in 1882 - a name usually attributed to a translation slip between the German words for "surface" (Fläche) and "bottle" (Flasche). Henri Poincaré formalized topology as its own branch of mathematics around 1895.
Main Contributors
Leonhard Euler, August Möbius, Johann Benedict Listing, Felix Klein, Henri Poincaré, and in the 20th century William Thurston, who reshaped low-dimensional topology and geometry.
Latest Trends
Real-time WebGL/Three.js rendering makes previously abstract surfaces directly manipulable in a browser rather than confined to a diagram. Knot theory has growing ties to biology (how DNA knots itself) and quantum computing (topological qubits). And generative art is starting to apply fractal deformation to classical topological surfaces - a "fractal Möbius" or "fractal torus" - which is the direction this site's own roadmap plans to take the category next.