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In this programme it is shown that the parallel axiom is consistent with and independent of the other Euclidean axioms. In order to show this it is necessary to construct a universe in which normal... concepts about lines and triangles do not appear to operate.
Module code and title: M101, Mathematics: a foundation course M101; 28 03-09-1978 1978 00:25:00 + Show more... Jack Koumi Colin Rourke; Allan I..,1936-2013 Solomon BBC Open University Axioms; Poincare Allan Solomon introduces the programme and then Colin Rourke looks at the five groups of axioms used in Euclidean geometry, including the parallel axiom. An animated diagram shows how this axiom could be constructed and also misinterpreted, for example if the lines lay on a curved surface. Colin Rourke demonstrates that the parallel axiom, although consistent with the incident axioms, is independent of them. He does this using two models, one of which satisfies all the axioms involved, and a second model which satisfies the incidence axioms but not the parallel axiom. Allan Solomon now attempts to prove the consistency and independence of the parallel axiom with the complete set of Euclidean axioms. The Cartesian plane represents the model which satisfies all the axioms. The second model is a Universe in which everything contracts as it moves from the centre. This is represented by trick photography. Colin Rourke is projected inside this world which from the inside appears normal. Allan Solomon now demonstrates that within this universe there are no unique parallel lines through a point, even though it may appear that there are to a person inside this world. All the other Euclidean axioms are shown to hold in this world. The parallel axiom is therefore consistent with, yet independent of, the other Euclidean axioms. A further insight into this world is given when it is shown that the angles of a triangle in this world add up to 150 degrees . Allan Solomon concludes the programme by observing that we cannot resolve the problem of whether or not we live in a similar universe. 6HT/72732 00525_4266 499 no