Robert Dixon

THE MATHEMATICS AND COMPUTER GRAPHICS OF SPIRALS IN PLANTS

This is a study of the pattern known as spiral phyllotaxis—literally, “leaf-arrangement”, but applied to the arrangement of seeds, florets, petals, scales, twigs and so on—and which is very widespread in plants. A number of stages in the refinement of model-making are recorded. In particular, the model illustrated in the literature on the subject indicating uniform growth is taken a step further in generality to indicate non-uniform growth. The intention has been to arrive at a wholly visual statement. And yet the mathematical approach is integral to this end. The study of flower forms becomes a study of the relation between the circle and the golden ratio. A computerised drawing system was used to draw forms that are algebraically defined. This solved a technical problem, but also gave me a glimpse of the enormous scope opened up by this electronic tool. To draw, for example, a simple equi-angular spiral (as found in snail shells) is difficult and time-consuming by traditional means, but elementary by computer.

Aesthetics: A Cognitive Account

This essay recalls the original meaning of aesthetic as relating to objects and acts of sense perception as opposed to objects and acts of formal knowledge. This distinction is then applied to the problem of defining art, so as to provide an approach along fundamental cognitive lines. Contemporary trends in the teaching of mathematics, which often also isolate sense perception from formal knowledge, are reviewed. New developments in mathematics which re-emphasize the importance of sensory perception to mathematics are then examined. The author argues that a unifying approach to aesthetics, art and mathematics is called for.

Two Conformal Mappings

Following his discussion of the classificational place of conformai mappings among elementary transformations, the author describes and illustrates two particular conformai mappings, inversion and anti Mercator, in order to draw attention to the symmetry implications of their combined use.