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The programme illustrates two examples of non-Euclidean geometry and uses Euclid's parallel postulate as a starting point to investigate these, other geometries. The experimental confirmation of Ei...nstein's theory of special relativity in an experiment in 1919 showed that the Universe itself was non-Euclidean even though the effect was very small.
Module code and title: U202, Inquiry U202; 03 01-04-1981 1981 00:24:00 + Show more... Tony Jolly Fred C. Holroyd; Colin Rourke; Allan I.,1936-2013 Solomon BBC Open University Euclidean geometry(non); Geometry; Parallel postulate; Relativity The programme begins with Allan Soloman and Fred Holroyd playing chess. Fred reveals that he is playing a variation on the rules of chess, cylindrical chess. The two of them discuss the differences this development would make to a game of chess. Allan then relates this introduction to the ideas of Non-Euclidean Geometry. He argues that a mathematical system is just like a game, the rules can be changed and different results will be obtained. Fred Holroyd argues that Euclid's geometry can be restructured by altering a rule or postulate, the parallel postulate, which need not be true at all. He describes what the parallel postulate says. Take a line and a point not lying on that line then there is one line only that you can draw through that point, parallel to the first line Allan Soloman sets out to challenge this statement by drawing a line on a sphere with a curved ruler. This generates a discussion into the definition of a straight line on a flat surface and on a sphere. After some discussion Allan Soloman demonstrates a property of straightness that applies in both cases. Using a ruler he draws a line, he then turns the ruler so that the edge is lined up with the other side of the line. They match perfectly. Fred does the same test with a bent ruler, but one which still lies flat in the plane. When the ruler is moved the line and the edge no longer match up. Allan and Fred conduct this test on a sphere with a curved ruler. This gives a straight line. These lines however come to a point at both ends of the sphere. Allan argues that one cannot have parallel straight lines on a spherical surface, but only parallel curves. He demonstrates this to Fred. Allan states that this is a case where our intuition goes wrong. Allan Soloman introduces a clip from M101/TV28, in which Colin Rourke is shown inside a non-Euclidean world, where everything contracts as it moves from the centre of the world. What seems like a straight line between two points to us appears to Colin Rourke as a curved line and vice versa. Allan explains why this should be so. Fred Holroyd discusses the geometry of Colin's world, showing how a triangle changes shape and size as it is moved from the centre. He argues that this world is another where the normal rules of geometry that we perceive intuitively, don't hold. He then shows how it would be possible to mistake a 3-dimensional world as one which only contained 2 dimensions. This means that Euclid's five axioms aren't logically necessary or self-evident. He argues that our universe could be four dimensional, or higher and curved. Allan Soloman argues that this inquiry into Euclid's parallel postulate has led to unexpected results. These have enhanced Einstein's theory of relativity which predicts that on the scale of the Universe, space is non-Euclidean. This was proved in an experiment in 1919 during a total solar eclipse. Finally, Allan sums up the relationship between mathematical inquiry and its relevance to the real world. 6HT/73469 FOUD116D 1299 no