Homotopies are studied in a branch of mathematics known as Algebraic Topology. A homotopy is a deformation (change of form) of one thing into another without cutting it. For example, if we imagine a stretchy object, then all the shapes we can stretch or twist it into are "homotopy equivalent". One famous example is the homotopy equivalence of a coffee cup and a donut. To a topologist, these two shapes are the same.