The famous physicist Sir Roger Penrose called the Hopf fibration “an element of the architecture of our world” – reason enough to try and visualize this remarkable map. Through the formalism of quaternions, S3 (the sphere in four dimensions) can be identified with the set of rotations of S2 (the spere in three dimensions). The Hopf fibration now maps a point r from the S3 to the point P on the S2, to which the point (1,0,0) gets rotatated by the rotation defined by r. It turns out that in this construction, the preimage of a point P on S2, called the fiber of P under the Hopf map, is a circle on S3. Being humans, we unfortunately cannot see the structure emerging from this in 4 dimensions. But instead, we can use stereographic projection to construct a visible image of the Hopf fibers. As different circular fibres on S3 are interlinked and stay interlinked circles in R3 under projection, beautiful structures emerge by looking at the Hopf fibrations of curves on the sphere. The figures show different visualisations of the Hopf fibration of a spiral on the sphere.