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Description
In this programme the group of isometrics is used to study space. It is shown that any isometry can be obtained through a combination of three transformations, rotation, translation and reflection.... The sub-group of symmetry is used to explore the isometry group in more detail.
Module code and title: M203, Introduction to pure mathematics M203; 01 16-05-1978 14-02-1980 1980 00:24:30 + Show more... Jack Koumi Colin Rourke; Allan I.,1936-2013 Solomon Open University. group; Isometry; translation Allan Solomon introduces the programme, explaining that the course will concentrate upon the mathematics of space. Colin Rourke discusses the concepts of space and motions in space, using the movement of a bus in a garage as an example. He defines space as all the places where an object can exist and be moved around isometrically. Allan Solomon then discusses some problems connected with this definition. Having surmounted these problems he then goes on to ask what are the properties of isometries. Colin Rourke argues that they form a transformation group, satisfying three properties, closure, identity and finally inversion. Allan Solomon argues that M203 is studying ordinary space and distance as a group. Different geometries belong in other groups, with other allowable motions. Ordinary Euclidean geometry is defined by the group being studied in this programme. Colin Rourke defines three types of isometry using Euclidean concepts. Translation, Rotations and reflection. He uses models to help to fully explain these terms. Allan Solomon then explains that any isometry can be accomplished by using a combination of these three isometries. Colin Rourke now attempts to prove that statement by showing that any isometry can be reproduced on the four vertices of a tetrahedron by use of the three isometries. This result then applies for all points in space, Colin and Allan demonstrate this to be so. Allan Solomon now proposes the study the isometry group by looking at a sub-group, the symmetry group of an object. Colin Rourke examines the symmetries of a tetrahedron with an equilateral triangle as a base and the other three faces isosceles triangles. He deduces the number of elements in the symmetry subgroup for this tetrahedron by rotating about axes passing through vertices and reflecting in planes passing through edges. Allan Solomon now sets the task of finding the number of symmetries for a more regular tetrahedron, where every face is an equilateral triangle. Colin Rourke argues that there should be twenty four symmetries for such a body. He sets about demonstrating these by rotations about axes passing through vertices and reflections in planes passing through edges, giving nine new symmetries over and above the ones originally discovered. He then finds three more symmetries by rotating through axes passing through the mid points of edges. He then obtains a further six symmetries by a combination of rotation and reflection. Allan Solomon explains this a little further and then sums up the findings of the programme and its relation to the rest of the course. 6HT/72890 FOU918J 833 no