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Computer animation and a 3-D model of a perspectivity demonstrate that Euclidean, Non-Euclidean and Protective geometries are unified and also distinguished by Klein's view of each one as a space p...lus a group of transformations. For each geometry, there's a particular set of properties of figures which are "essential" - and the geometry's transformation-group preserves just these properties. In terms of these properties, a new geometry, Affine geometry, is intermediate between Euclidean and Protective geometries: a model of an Affine transformation as a succession of parallel projections between planes, shows that ratios of lengths along a line are preserved (unlike Protective) while distances are not preserved (unlike Euclidean). This sequence of successive generalisations of a geometry is taken up in the second half of the programme in terms of group inclusions: animations and models are used to prove that the Euclidean Group is a subgroup of the Affine Group, which in turn is a subgroup of the Protective Group. The latter inclusion is demonstrated by showing that any parallel projection can be achieved by a sequence of two perspectivities.
Metadata describing this Open University video programme
Module code and title: M203, Introduction to pure mathematics
Item code: M203; 25
Recording date: 14-02-1979
First transmission date: 1980
Published: 1980
Rights Statement:
Restrictions on use:
Duration: 00:24:16
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Producer: Jack Koumi
Contributors: David Crowe; Jeremy Gray; Allan I.,1936-2013 Solomon
Publisher: BBC Open University
Keyword(s): Perspectivity; Non-Euclidean geometry; Euclidean geometry; Protectivity geometry
Master spool number: HOU3128
Production number: FOUM017Y
Videofinder number: 847
Available to public: no