When considering ‘convergence’ many people think of number sequences or contexts arising from calculus. But there are also interesting phenomena of convergence – easy to visualize – arising in elementary geometry. Some of them are so elementary that they can be dealt with at school, for instance an example of iteration that is described in [1, p. 171f] (see also [2, p. 59], [3, p. 222f] and [4, p. 42ff]).

The cover picture shows a 22° halo seen in Preston on Sunday 28 October 2001 at 10 am. In contrast with a rainbow, the red band is on the inside rather than the outside, the colour bands are much thinner, and the arc is centred round the sun instead of being opposite to the sun. Also, it is not raining. Haloes occur on still sunny days when there is plenty of cirrus cloud. It is possible to see a full circle depending on how high the sun is in the sky. This never happens with a rainbow except when seen from an aeroplane. Sometimes the colours are particularly vivid at the 3 o’clock and 9 o’clock points. These vivid areas are called parhelia or mock suns. Sometimes the parhelia are all that can be seen.

Imagine a horizontal rectangular tray with vertical sides, and a disc lying flat in the tray and rolling around the inside of the tray in such a way that its circumference is always in contact with the edge of the tray. We shall suppose that the disc has been rolling for ever in the past, and that it will continue to roll for ever in the future. What can we say about this physical system? Our aim is to illustrate the complexity and the discontinuous nature of the system under arbitrarily small changes in the size of the tray (a feature that is typical of many dynamical systems).