Natural History Magazine

How can it be that mathematics, being after all a product of human thought
which is independent of experience, is so admirably appropriate to the objects of reality?
—Albert Einstein, lecture before the Prussian Academy of Sciences, January 27, 1921.

Through mathematics one can enter a purely abstract world—one that I recently rediscovered while reading The Number Devil, by the German author Hans Magnus Enzensberger, to my children. With wonderful illustrations by Rotraut Susanne Berner, the book takes readers into the surreal dreams of a troubled math student who is visited nightly by an irritable teacher with a pointing cane, red skin, and horns. Together, student and teacher venture into territory rarely explored by the schools, which confine themselves to the materials covered by standardized tests. The Devil unveils a rich world beyond basic arithmetic. The numbers form curious patterns, almost as if they were alive. They create their own reality.

The big surprise is that even the most arcane realities of abstract mathematics often end up offering deep insights into the natural world. In 1960 the physicist Eugene P. Wigner published his classic paper, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Going beyond his title, Wigner makes the point that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.” Einstein’s famous formula E=mc² is just one example of how the natural world can be neatly reduced to equations. Wigner was hopeful that the more complex biological realm would eventually yield to mathematical description as well.

Before looking at Web sites about the mathematics of biology, however, it’s useful to contrast the complicated “messiness” of biology with the relative order of the inorganic world. For me, the most familiar examples of orderly nature come from mineralogy—indeed, it’s hard to imagine anything in nature more regular than the geometry of a cubic grain of salt or a six-sided prism of crystalline quartz. Undoubtedly, early Greek mathematicians marveled at the perfection of crystals, but the math explaining their outward appearance—a result of the regular spacing of atoms—came much later, in a branch of mathematics dealing with the packing of spheres. The first to explore the question of optimal packing of spheres were such great mathematicians as Johannes Kepler and Carl Friedrich Gauss.

At a French Web site, software designer Marcus Hewat has a page called Making Matter that illustrates how atoms assemble into remarkable geometries.

To examine the beautiful geometry of the ways atoms are packed into more than 3,500 different minerals, go to the Webmineral page. There you will find a menu with a number of Java files that enable you to rotate each crystal lattice in space.

Another page, offered by the makers of the Polymorf construction-set toys, explores the geometry of crystal structure.

Some of the earliest mathematical patterns observed in the living world were the perfect spirals that occur in pine cones and flowers. Ron D. Knott, a former lecturer at the University of Surrey in Britain, maintains an extensive site on Fibonacci numbers, named after Leonardo Fibonacci, perhaps the most talented mathematician of the European Middle Ages. Scroll down to “Fibonacci Numbers and Nature” to learn how this sequence of numbers appears repeatedly in nature. In other sections of his site, Knott shows how the golden section and golden string—mathematical concepts related to Fibonacci numbers—also reveal themselves in nature.

On another British Web site, this one maintained by a dentist named Eddy Levin, who practices on fashionable Harley Street in London, specially designed calipers demonstrate that The Golden Proportion can be found everywhere. Levin has even used the calipers in his practice to improve the appearance of dental reconstructions. Click on “Nature” in the menu to see how the calipers can be applied to natural patterns ranging from peacock feathers to the stripes on a ring-tailed lemur.

Phyllotaxis is a beautifully designed site that explores the spiral patterns that occur in plants. The site was developed by students in the math department of Smith College, in conjunction with the botanic garden at the college. Click on “Gallery” to see how the spirals appear in cactuses and other plants, or go to “Applets” to play with interactive demonstrations of the underlying principles.

Animals, too, display inherent mathematical patterns. One page at the Web site of the American Mathematical Society features The Mathematical Study of Mollusk Shells. Click on section six to find a Java applet that enables you to specify the basic shape of just about any snail shell by adjusting three just variables. In 1962 David Raup, a paleontologist then at Cornell University, outlined the use of the three variables, one of the first efforts to model animal forms with a computer.

Øyvind Hammer, an assistant professor at the Natural History Museum at the University of Oslo, Norway, has a page on Computational Paleontology, which offers a few examples of mathematics from deep time. I was particularly intrigued by the suture lines from extinct mollusks called ammonites, which conform to a kind of fractal called a Koch curve.

At Plus magazine, Lewis Dartnell, a biologist of University College London, writes about How the Leopard Got Its Spots. In his article, Dartnell revisits the one paper by Alan Turing that has any bearing on biology. Turing was the legendary English mathematician who helped crack the German Enigma code and devised many of the principles of the digital computer. Turing’s work half a century ago anticipated work in developmental biology that shows how patterns emerge on animals of all kinds.

A site maintained by Derek Locke in England has an eclectic assortment of pages on mathematics and nature. In addition to Fibonacci numbers, he delves into objects such as spiderwebs, which are mined with sticky blobs conforming to a mathematical curve called an unduloid. He also looks at the mathematics of snake locomotion.

Evolution, a rather messy and complicated affair that unfolds over thousands of generations, has not lent itself to simple mathematical analysis. Nevertheless, its primary mechanism, natural selection, can be crudely imitated with a computer algorithm. Barry G. Adams, a mathematician at Laurentian University in Sudbury, Ontario, Canada, has a site that runs a simulation first proposed by the evolutionary biologist Richard Dawkins in his book The Blind Watchmaker. The math is unlike anything taught in grade school; each time you run the experiment you are likely to get a different answer (something the Number Devil would love). Starting with a string of randomly chosen letters and spaces, the program runs, selecting for matches with Dawkins’s chosen line from Shakespeare’s Hamlet, “Methinks it is like a weasel.” If you select a large number of “children” per generation (each “child” has only a single-letter change from the “parent” string), say 100, the string of gibberish quickly “evolves” a match within fifty to sixty generations—a lot faster than random chance.

Finally, in a recent feature at the Web site of the American Mathematical Society, David Austin, a mathematician at Grand Valley State University in Michigan, writes about current efforts to find patterns in the loops of DNA. Austin’s aim is to help unravel the workings of evolution on a microscopic scale. Titled That Knotty DNA, Austin’s article shows how the study of knots (another branch of mathematics never mentioned in grade school) is being applied to model the tangling and untangling of DNA in the nucleus.