Description
This programme looks at the many 'classes' of topological surfaces, all of which are based on just three characteristics: number of boundaries, Euler characteristic and orientability.
Module code and title: M335, Studies in pure mathematics M335; 01 13-02-1983 1983 00:24:20 + Show more... Jack Koumi Jeremy Gray; John Peters BBC Open University Boundaries; Euler characteristic; Surface; Topological class Over shots of a torus which has on it a representation of the continents and of shots of a globe, Jeremy Gray introduces the programme. He points out that the torus is intuitively not like the sphere and that this 'crude' classification enables an enormous variety of surfaces to be reduced to a manageable set of classes. Shots of various shapes rotating on a turntable. John Peters goes on to define a topological surface. He demonstrates this by pointing to a small area of a torus and explains that any surface is locally disc like, metric and compact. Shots of cylinders rotating on a stand. Over shots of various topological shapes rotating on a turntable, Jeremy Gray explains the criteria by which surfaces are lumped together into single topological classes. This leads him to discuss homeomorphisms - the relationship between two surfaces which are topologically the same (a one to one map from one surface to another which is contiguous). John Peters cuts up a torus to demonstrate the basic techniques for studying complex surfaces. He cuts the torus and flattens it to a rectangle. Peters explains that the rectangle is topologically the same as a torus and that a torus can be studied by examining a rectangle. Peters next cuts up a sphere which has on it a representation of the earth's continents. He points out the importance of identifying the edges for reconstruction and also that there are several ways in which surfaces can be cut and flattened. Peters then explains, briefly, that even very complicated surfaces can be subdivided into polygons for study in the same way as simple surfaces. Jeremy Gray introduces the concept of the Euler characteristic which is used to give a numerical value to the characteristic of surfaces. Gray goes on to test the Euler theorem on several surfaces and their flat representations. Although the theorem isn't proved in this programme, Gray does show that it holds for all his examples. John Peters explains that the Euler characteristic can be used not only to tell that one surface is different from another, but also how different it is. He uses models of spheres and toruses to illustrate his points. To end the programme, Jeremy Gray discusses, briefly, how a classification of surfaces can be built up. He points to several topological models as he talks . HOU4004 FOUM131S 973 no