On the intrinsic elegance of figures with divine proportions


August 18, 2000

Self-similarity pervades nature. The logarithmic spiral, for instance, may be observed in sunflowers, snails, sea-shells and sheep's horns. If the end of such a spiral is deleted, the remainder may be photo-enlarged and rotated to coincide exactly with the original. This is an example of a "gnomon", a term used by Hero of Alexandria for a form that, when added to a given form, results in one similar to the original.

The beauty of self-similarity, or "gnomonicity", in nature and art may be apparent to all, but it falls to mathematics to describe and analyse the underlying regularity. Gnomonicity is reflected in the mathematics itself - in number patterns, in formulae and in geometry.

The sequence of squares - 0, 1, 4, 9, 16, 25... - provides a simple example.

The gnomons, obtained by subtracting successive terms, are just the odd numbers 1, 3, 5, 7, 9.... These gnomons have a geometric interpretation: starting with a 4 x 4 square array of dots, adding 9 dots (4 on the right, 4 along the bottom and 1 in the corner) yields a 5 x 5 square array, and so on.

Gnomon: From Pharaohs to Fractals rapidly progresses from such basic arithmetic examples to continued fractions, which are central to the understanding of many of the gnomons encountered later. Continued fractions are a method of representing numbers, something like a decimal expansion, but in an especially efficient way. Their gnomonicity is a consequence of their recursive definition, but is also apparent in their typographical form. For example, the square root of 3 has continued fraction expansion 1+1/(2+1/ (1+1/(2+1/(1+1/(2+...))))); removing initial terms leaves a similar expression. The simplest continued fraction 1+1/(1+1/(1+1/(1+1/ (1+...)))) is the "golden number" or "divine proportion", 1.618I , a number long associated with intrinsic elegance in art, music and nature, but which is also fundamental to mathematics and geometry. The richness of such continued fractions and the closely related Fibonacci sequences (sequences where each term is the sum of the preceding two) yields a myriad of attractive inter-relationships and gnomonic formulae.

The book has a geometric emphasis, providing visual analogues of many of the more formal mathematical gnomons. For example, "whorled figures", obtained by building outwardly spiralling sequences of abutting squares or rectangles, allow a pictorial representation of Fibonacci sequences and continued fractions. Gnomonic spirals are examined in their own right, leading in turn to the mathematical gnomonicity of rotation matrices.The fractal constructions of the final chapter display an even richer visual self-similarity, with copies of a generating figure apparent throughout the fractal and at many scales.

From pure mathematics to applied: a chapter on electrical networks and mechanics presents striking physical analogues. Continued fractions are displayed side by side with transducer ladders, resistance ladders and pulley systems, which emphasise mathematical parallels between apparently diverse structures, parallels that deserve to be more widely known.

Gnomon: From Pharaohs to Fractals is primarily a mathematics book, despite plenty of pictorial examples and occasional historical interludes.

The paucity of references leaves the origin and evolution of many of the ideas and their interpretation unclear. The book assumes a mathematical knowledge that, at least in Britain, is now left until degree courses, but it also requires a maturity to appreciate elegant mathematics in a novel context. Considerable dedication is required to penetrate the concentrated pages of, albeit attractive, gnomonic formulae on Fibonacci sequences.

Again, although the electrical theory is introduced from scratch, readers unfamiliar with electrical engineering will struggle with the conceptual sophistication of imaginary impedance, discretisations of transmission lines and wave propagation.

Who will read the book? It is a recreational mathematics book, but for the accomplished mathematician. The author acknowledges the influence of Martin Gardner and Ian Stewart, but most of the popular audience who have enjoyed their classics will struggle here. Adapting a suggestion by Petr Beckman in one of the chapter-head quotations, much of the book concerns the "lost mathematics" that is now too advanced for school but too offbeat for a university course. Perhaps the book will appeal most to those who have studied advanced mathematics in the past and wish to return to some of the gems that they may have missed.

The book follows an enthusiastic, if idiosyncratic journey among topics that fascinate the author; a fascination of the structure and symmetry of individual gnomons, and of their endless web of inter-connections. Nevertheless, while penning an account of such a journey may be highly satisfying for the author, it may be less appealing to those for whom the discovery is second-hand.

Kenneth J. Falconer is professor in pure mathematics, University of St Andrews.

Gnomon: From Pharaohs to Fractals

Author - Midhat J. Gazale
ISBN - 0 691 00515 1
Publisher - Princeton University Press
Price - £18.95
Pages - 259

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