The Hopf fibration (left) is a complex shape discovered in the field of topology that resembles a series of linked rings wrapped into a torus. Experimentalists used a laser beam to reorient the molecules in liquid crystals such that their molecular field—represented by lines that correspond to one specific orientation of the molecules in space—formed the same shape. (Center) A simulation of what the Hopf fibration should look like using the authors’ technique for mapping the liquid-crystal director field to a color map. To read the visualization, note that every point on the surface of the torus lies in the xy plane, while the colors correspond to points on a color wheel and define the polar angle of the director field in the xy plane. (Right) The winding of the actual field lines, as measured with fluorescence microscopy.