Available online
Description
Animations introduce three problems concerning Shrinking Polygons. The first problem is that of taking a unit circle and inscribing an equilateral triangle inside it then inscribing a circle inside... the triangle and further inscribing a triangle inside the circle and so on ... Repeat the sequence infinitely many times and the programme proves that the radii of the circles tend to zero. The second problem is similar except the inscribed regular polygons are first an equilateral triangle, then a hexagon then a dodecagon and so on ... Using a half-angle formula 'trick' attributable to Euler it is proved that the limit of radii is exactly 3 root 3 divided by 4 pi. The third problem is similar again except the inscribed regular polygons are first an equilateral triangle, then a square, then a pentagon and so on. By applying the Monotone Convergence Theorem the programme proves at the radii of the circles don't tend to Zero because they are certainly always greater than 10. A fourth section of the programme looks in detail at one of the results assumed when solving the second problem. The programme ends with a brief look at a model of the platonic solids and how these first prompted Kepler to consider the problems of shrinking polygons. (This programme was re-edited in 1995 as FOUM479P)
Animations introduce three problems concerning Shrinking Polygons. The first problem is that of taking a unit circle and inscribing an equilateral triangle inside it then inscribing a circle inside... the triangle and further inscribing a triangle inside the circle and so on ... Repeat the sequence infinitely many times and the programme proves that the radii of the circles tend to zero. The second problem is similar except the inscribed regular polygons are first an equilateral triangle, then a hexagon then a dodecagon and so on ... Using a half-angle formula 'trick' attributable to Euler it is proved that the limit of radii is exactly 3 root 3 divided by 4 pi. The third problem is similar again except the inscribed regular polygons are first an equilateral triangle, then a square, then a pentagon and so on. By applying the Monotone Convergence Theorem the programme proves at the radii of the circles don't tend to Zero because they are certainly always greater than 10. A fourth section of the programme looks in detail at one of the results assumed when solving the second problem. The programme ends with a brief look at a model of the platonic solids and how these first prompted Kepler to consider the problems of shrinking polygons. (This programme was re-edited in 1995 as FOUM479P)
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 05A; 1987 |
| First transmission date: | 1987 |
| Published: | 1987 |
| Rights Statement: | Rights owned or controlled by The Open University |
| Restrictions on use: | This material can be used in accordance with The Open University conditions of use. A link to the conditions can be found at the bottom of all OUDA web pages. |
| Duration: | 00:24:43 |
| + Show more... | |
| Producer: | Michael Peet |
| Contributors: | Phil Rippon; Allan I.,1936-2013 Solomon |
| Publisher: | BBC Open University |
| Keyword(s): | Animations; Euler; Half angle formulae; Kepler; Limit; Monotone Convergence Theorem; Null Sequences; Platonic solids; Regular Polygons; Squeeze rule |
| Subject terms: | Numbers, Polygonal; Polygons |
| Master spool number: | HOU5696 |
| Production number: | FOUM272D |
| Available to public: | no |
