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Description
This programme deals with patterns or flows on surfaces by analogy with fluid flow over a surface.
Module code and title: M335, Studies in pure mathematics M335; 08 04-09-1983 1983 00:24:22 + Show more... Jack Koumi Graham Crowe; Flegg David BBC Open University Flows; Hairy surfaces; Index sum; Singularity; Sphere; Topological difference; Torus What sorts of pattern can you make on a hairy sphere if you brush the hairs smoothly? What about a hairy torus? For example you can brush a sphere smoothly along its meridians and end up with a crown at the north pole and a tuft at the south pole. Can you get a similar pattern on the torus? Such patterns are called "flows" (by analogy with fluid flow over a surface). In the mathematical definition of a flow, there's a speed and direction at each point of the surface; but at some points, called "singularities", the speed is zero and the direction is indeterminate (e.g. at the crown or the tuft, above). However, different types of singularity can be distinguished by the pattern of the flow which surrounds them. The programme uses several animations to show how each such pattern can be assigned a characteristic value, called the "index" of the singularity. Several flows are demonstrated on hairy surfaces, until it begins to look as if no flow on the sphere could have the same index sum as a flow on the torus. However, all the flows demonstrated on the sphere have the same index sum: two; and all those demonstrate on the torus have index sum zero. The last half of the programme uses a variety of animations to show that this is a genuine topological difference: for each surface, if you calculate the index sum for any flow, you'll always get the same answer, moreover, this turns out to be the Euler Characteristic of the surface. HOU4208 FOUM154H 980 no