Fibration
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Lesson Summary:
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.
Lesson Description:
Complex numbers are used to create beautiful arrangements of circles in space.
Questions answered by this video:
- Who is Heinz Hopf?
- What is fibration?
- What is topology?
- What does a fibration look like?
- How can a 3-sphere be projected onto a 2-sphere?
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Staff Review
Arrangements of circles in space are discussed in this beautiful video on fibration. This video takes some of its ideas from previous videos in the series on complex numbers and the third and fourth dimensions. Complex planes and the S3 sphere are shown and explained in four dimensions. Stereographic projections are again used in this lesson.
