The Perfect Shape: Spiral Stories

Oyvind Hammer works at the Natural History Museum at the University of Oslo in Norway. He has produced a curious book that is full of short segments on spirals, their mathematics and their occurrence in both the natural and man-made worlds. In fact, this book constitutes one of the wildest rides through spirallography outside of D’Arcy Wentworth Thompson’s On Growth and Form. It is replete with fascinating pictures (prehistoric ammonites, a spiral water pump, stereo amp, unrolling ferns, gnomons) and some very cogent mathematics. In fact, I would argue that this is one of the few such copiously decorated books that is actually faithful to the mathematics behind the logarithmic spiral.

Hammer takes us back to Descartes the mathematician (and not the guy who is “famous for less important contributions, such as being because he thought, and dying from a cold he picked up in Sweden”, p. 33). In September 12 and October 11 of the same year 1638, Descartes wrote to Father Mersenne and discussed the two most important properties of the logarithmic spiral: (a) the total arc length from the origin along the spiral to a point C on it divided by the radius length AC is constant; (b) the angle between the radius and tangent vector is constant (the equiangular nature of this spiral). The proof that each of these uniquely characterize such a spiral is left to an appendix and a modern calculus computation. The entirety of the book is filled with such nuggets (and, I might add, moments of rigor). Yes, it really does deserve to be published by Springer. It is a worthy addition, in fact, to the Copernicus imprint of popular books.

So what’s in the rest of the book? Loxodromes, several discussions of chirality of spirals and snails, snakes which lie in the deepest parts of the ocean, the spiral casing for the Francis turbine, circular tessellations in Roman mosaics domes of mosques and the pattern of “eyes” in a peacock’s tail feathers. And so much more! Consider Chapter 23, “Thou Shalt Love Thy Neighbor”: amorous mice are positioned at the vertices of a square and pursue each other. What curve results? The logarithmic spiral of course. Or Chapter 20 “How to grab a can of beer” in which someone’s hand opens to form a circular arch. And the path of each fingertip is? You guessed it. A spiral but not logarithmic. Or Chapter 26 “The Killer Spiral” about bending of membranes in bimetals, pine cones and cinnamon sticks (in all cases driven by either temperature changes or dehydration).

Of course plenty of history is involved as well. Chapter 29 “Newton’s Spiral Headache” details the many arguments between Johann Bernoulli and Isaac Newton over the shape of orbits. Bernoulli disagreed with Newton over the necessity of conic section orbits under an inverse square law. Bernoulli was interested to see if he could produce other simple curves (including spirals) under the influence of other radial force laws (such as an inverse cube law). The argument was clarified by Roger Cotes (Harmonia Mensurarum, 1714) who found three other spirals resulting from an inverse cube law: the secant spiral, the hyperbolic secant spiral and the hyperbolic cosecant spiral).

Many chapters contain mathematical modeling arguments and calculus computations for such things as seashell development or propulsion or packing of florets in a flower head . A brilliant example of such modeling is contained in my favorite chapter “Coffee, Kepler and Crime”. In this thought-experiment a cup of coffee is stirred in a steady manner and then a straight stripe of milk is added from center to wall of cup. Should the stripe deform into a logarithmic spiral? Should this same experimental setup also apply to the formation of spiral galaxies? I’ll leave this as an exercise for the reader. This book would not be complete without featuring Archimedes work on spirals and this does occur in a number of places.

The mathematically-oriented reader will likely be fascinated by this charming book. It is full of digressions on so many topics that one’s head will likely be spinning after reading a few chapters. Ah, but what curve will fit that delightful spin? One wonders.

Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.