Available online
Description
This programme examines the work of Felix Klein, in which geometries are classified into groups, determined by the 'invariants' within each geometry.
This programme examines the work of Felix Klein, in which geometries are classified into groups, determined by the 'invariants' within each geometry.
| Module code and title: | M101, Mathematics: a foundation course |
|---|---|
| Item code: | M101; 29 |
| First transmission date: | 17-09-1978 |
| Published: | 1978 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:25:00 |
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| Producer: | John Jaworski |
| Contributors: | Bob Margolis; Allan I.,1936-2013 Solomon |
| Publisher: | BBC Open University |
| Keyword(s): | Affine geometry; Erlangen Programme; Euclidean geometry; Group theory |
| Footage description: | The programme begins with Allan Solomon explaining that he wants a general proof for the proposition that the lines through the mid points of the lines of a triangle will all pass through the same point. In order to show this he first wants to look at Klein's work on cataloguing geometries. Bob Margolis looks at the properties of some non-Euclidean geometries. Allan then discusses Klein's approach which involves the study of invariants within these geometries. Bob Margolis now looks at Euclidean geometry and isolates its invariants, which are shape and size. Any transformations within this geometry involve the whole plane, which is moved rigidly. He then shows that these transformations constitute a group. Allan now shows that in every case where transformations of a plane are considered, there is a group. This enables us to build up a catalogue of geometries. Bob examines the variants in Similarity geometry, where shape is fixed, but size and position can be altered. This is seen to include the earlier Euclidean group, and is included in the hierarchy of geometries. Protective geometry is now considered, where shapes change, but linearity is preserved and triangles are mapped to triangles. This group is demonstrated by moving a triangle through a pointed light source and observing the image cast upon a screen. The midpoints of these triangles are not preserved however. When this light source is replaced by a parallel light source we then get a further group, affine geometry in which mid points are preserved. This group can now be used to prove the proposition discovered at the beginning of the programme. Allan Solomon places these two new groups into the hierarchy and then discusses the usefulness and importance of Klein's approach to geometry. |
| Master spool number: | 6HT/72790 |
| Production number: | 00525_4267 |
| Videofinder number: | 449 |
| Available to public: | no |
