The Hopf fibration

The small sphere at the top right is an ordinary sphere, and each of its points stands for one circle in the three-sphere, a sphere in four-dimensional space; projected into our space, it is the circle of the same colour in the main picture. Drag the three points on the small sphere (only the half facing you is in reach; press outside it and the point lands on its rim) and the three circles grow, shrink and move: a point near the north pole gives the smallest circle, circles grow larger further south, and near the south pole they spill out of the picture; the south pole itself gives a straight line (the small sphere leans towards you, so you cannot reach it). Wherever the points go, the linking number of each of the three pairs in the corner is 1: every two circles are linked, exactly once. When two points coincide, or very nearly, they give the same circle, and that pair is not measured: the corner shows a dash. Under ‘The gold point's latitude’ choose ‘Show the corresponding circles’ and the circles of the points on that latitude appear together, lying on a torus, each winding once round it at a slant; drag the gold point south and the torus swells from a thin ring into a wide one round the axis.