Since ‘a body continues in its state of rest or of uniform motion in a straight line except in so far as it is compelled by forces to change that state’, the very first question that should be asked about walking is why any energy at all need be expended in maintaining fixed speed on level ground. With a wheeled vehicle remarkably little force is exerted for this once it is got going, and virtually all the energy is used to overcome resistive drag. But the body is not dragged, and there is little resistance presented by relaxed, healthy limbs.

We are just contemplating the purchase of a new car, and have been testdriving a number of competing models. A mirror fixed to the driver's door is nowadays standard equipment, and on most of the models this is attached near the front of the door, more or less in line with the hinges. But in the VW Golf it is set further back; a good feature, claimed the salesman, since it reduces the driver's “blind spot”.

I confess that it was not obvious to me that this is so. If true, then perhaps it would compensate for the inconvenience of having to turn one's head further round to look in the mirror. But mathematical investigation seemed to be called for.

Vector fields model movement over an area or a volume. For example they can be used to describe the flow of air over an aeroplane wing, or past a car as it moves, and once these models have been set up they can be adjusted to produce an efficient, aerodynamic shape.

This article is simply based on the idea of addition of vectors in the plane and its aim is to produce effective displays of vector fields on the BBC microcomputer. Using animation techniques the vector fields are represented by a flow of dots on the screen. The program has been designed so that flows can be easily combined, making it possible to manipulate them as you wish to produce your own ‘aerodynamic’ two-dimensional car designs, ship-shapes, or bridge supports.

Applications of convex coordinates are many and varied in mathematics today. Our first interest in convex coordinates grew out of their use in linear programming, e.g. [1]. That interest has remained quite lively as the study of convex coordinates continues to yield pleasing and sometimes unexpected results, e.g. [2,3]. Another name for these coordinates which may be more familiar to the reader is that of barycentric coordinates as originally given by the German mathematician Möbius. In this article we shall confine our attention to the Euclidean plane, but convex coordinates may be defined in Euclidean spaces of any dimension.

The theory of duality is well known as a useful tool in the study of convex three-dimensional polyhedra. For example, if we know the types of polyhedra that can be circumscribed about a sphere, duality provides an immediate answer to the question as to what types can be inscribed in a sphere. Theorems like Euler’s Theorem (V – E + F = 2 where V, E and F are the numbers of vertices, edges and faces of a convex polyhedron) are self-dual in the sense that duality simply interchanges the values of V and F.