In this lesson, mathematician Heinz Hopf explains how complex numbers can be used to create a beautiful arrangement of circles in space called a vibration. By filling a three-dimensional sphere in four-dimensional space with circles, each associated with a complex line, a complex plane of dimension two is formed. The circles, which appear linked when viewed in groups, are an example of a vibration, with a circle of dimension one above each point on the two-dimensional base sphere. Hopf's discovery of this decomposition of the sphere into circles is known as the Hopf Fibration and has become a fundamental object in topology.
Complex numbers are used to create beautiful arrangements of circles in space.