Watch paths morph into each other — the fundamental idea of algebraic topology
A homotopy between two paths f and g is a continuous family of paths H(s,t) where H(s,0)=f(s) and H(s,1)=g(s). Two paths are homotopic if they can be continuously deformed into each other. This defines equivalence classes — the fundamental group π₁ counts "essentially different" loops.