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Description
This programme defines congruency in non-Euclidean geometry and then proves that the area of a triangle in this space is proportioned to pi minus the angle sum of the triangle.
This programme defines congruency in non-Euclidean geometry and then proves that the area of a triangle in this space is proportioned to pi minus the angle sum of the triangle.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 22 |
| First transmission date: | 24-07-1981 |
| Published: | 1981 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:22:55 |
| + Show more... | |
| Producer: | David Saunders |
| Contributors: | Jeremy Gray; Allan I.,1936-2013 Solomon |
| Publisher: | BBC Open University |
| Keyword(s): | Euclidean; Kaleidoscope; Non-Euclidean reflections; Geometric transformation |
| Subject terms: | Geometry, Non-Euclidean |
| Footage description: | Allan Solomon introduces the programme by briefly recapping on the nature of non-Euclidean geometry as depicted in M101. Jeremy Gray demonstrates these principles with simple drawings, showing that there are an infinite number of parallel lines through a given point to a given line. He then discusses the problem of copying triangles within the non Euclidean world. He argues that what is needed to resolve this difficulty is a definition of allowable transformations in non-Euclidean geometry. Using computer graphics he considers reflection in Euclidean geometry, and then reflection in arcs of circles in non-Euclidean geometry. The reflection sends points to points and arcs of circles to arcs of circles. Using computer animations he demonstrates inversion in a circle, a transformations in Euclidean geometry which also has these properties. This shows that under such a transformation equal angles always go to equal angles. He then looks at inversion in a non-Euclidean line. Using animations he demonstrates how all transformations in Euclidean geometry can be expressed as combination of reflections. He demonstrates that this is also true in the non-Euclidean world. These results allow him to define congruency, or allowable transformations, in non-Euclidean geometry. In particular two triangles are congruent when they have corresponding angles equal. Allan Solomon now proves the theorem that the area of any triangle within the non-Euclidean space is proportioned to the angle sum of the triangle taken away from pi. |
| Master spool number: | 6HT/73081 |
| Production number: | FOUM019L |
| Videofinder number: | 845 |
| Available to public: | no |
