An animated explainer on homotopy groups. Discusses geometric intuition for π₁/π₂/π₃ using loopspaces of metric spaces. Includes an original, nonstandard visualization of the Hopf map. This is my submission for #SoME4 ( ). Niles Johnson's Hopf fibration animation: A general reference is Hatcher's Algebraic Topology text ( ~hatcher/AT/ ). See especially Hatcher's Proposition about metric spaces, about the hom/product adjunction, and all of Chapter 4 on homotopy groups. Ivanov-Mikhailov-Wu paper: 00:00 Introduction 02:03 π₁ 15:07 π₂ 36:29 π₃ 47:10 πₖ in general