Homotopy

This simulation shows how a circular loop can be smoothly deformed into a figure-eight. In topology, such transformations are homotopies — they preserve connectivity without tearing or gluing. Homotopy equivalence captures the idea that two shapes can be continuously morphed into one another, revealing deeper structure about the space they inhabit.

This flow visualizes two loops twisting around each other in continuous transformation. The smooth braiding illustrates homotopic motion — both loops remain path-connected, showing that their topological identity is unchanged. Such braids relate directly to the algebraic structure of the fundamental group.

A null-homotopy demonstrates that a loop can be continuously contracted to a point within a space. This tells us that the space is simply connected at that region — any closed path can shrink down without tearing. This animation reveals how the fundamental group of such a space contains only the identity element.