Free Nonplussed! by Julian Havil Page B

a pursuit curve results from taking several objects, each one acting as pursuer and quarry. For example, in figure 10.4 we imagine four spiders, each starting at a corner of a square and moving with equal, constant speed toward one another. In figure 10.5 lines have been drawn linkingsome of the positions of pursuer and quarry and the diagram becomes a work of mathematical art.

    Figure 10.4. Four spiders in mutual pursuit.

    Figure 10.5. Four spiders with some links.

    Figure 10.6. A logarithmic spiral.
    These last, artistically satisfying, examples trace self-similar curves known as logarithmic (or equiangular) spirals , whose polar equation is r = a e bθ for constants a and b . A typical example is shown in figure 10.6 . They were first studied in 1638 byRené Descartes but are most famously associated with Jakob Bernoulli, who developed many of their startling properties. So enamoured of them was he that he asked for one to be engraved on his tombstone with the phrase ‘Eadem mutata resurgo’ (‘I shall arise the same, though changed’); unfortunately, the stonemason seems to have been unduly challenged by the charge and produced a somewhat crude Archimedean Spiral (whose polar form is r = aθ ). An essential difference between the two spirals is that successive turnings of the Archimedean Spiral have a constant separation distance (of 2 πa ), whereas with the logarithmic spiral these distances are in geometric progression.
    Logarithmic spirals abound in nature: they are the paths along which insects approach a light source and hawks approach their prey, the shape of spiral galaxies (including our own Milky Way) and also of cyclones. In Book 1 of Principia Newton proved that if the universal law of gravitation had been an inverse cubic law, rather than our familiar square law, a possible orbit of the planets around the Sun would have been that of a logarithmic spiral.
    Logarithmic spirals are remarkable curves for very many reasons, and one will prove to be the second curve needed for the solution to our principal problem.
    Before we do so, it is impossible to ignore an amusing anecdote relating to the remarkable analytic number theorist G. H. Hardy, in which he posits an equation which represents an equiangular spiral, which is also a parabola, and a hyperbola.
    During his tenure of the Savilian Chair of Geometry at Oxford, he gave his presidential address to the Mathematical Association in 1925, under the title, ‘What is geometry?’, in the course of which he said with characteristic clarity:
    You might object…that geometry is, after all, the business of geometers, and that I know, and you know, and I know that you know, that I am not one; and that it is useless for me to try to tell you what geometry is, because I simply do not know. And here I am afraid that we are confronted with a regrettable but quite definite cleavage of opinion. I do not claim to know any geometry, but I do claim to understand quite clearly what geometry is.
    He had, however, contributed to the geometrical literature with the following note, published in the Mathematical Gazette in 1907.
    224 . [M 1 .8.g.] A curious imaginary curve.
    The curve ( x + i y ) 2 = λ( x − i y ) is (i) a parabola, (ii) a rectangular hyperbola, and (iii) an equiangular spiral. The first two statements are evidently true. The polar equation is

    the equation of an equiangular spiral. The intrinsic equation is easily found to be ρ = 3i s .
    It is instructive (i) to show that the equation of any curve which is both a parabola and a rectangular hyperbola can be put in the form given above, or in the form

    and (ii) to determine the intrinsic equation directly from one of the latter forms of the Cartesian equation.
    G. H. Hardy
    We will not pursue his argument fully, partly as we have no wish to delve much into complex numbers here, but the ‘evidently true’ part of the statement seems to rely on two substitutions of variables:
    • X = x - i y and Y = x + i