D’Arcy Thompson (1860–1948) was a Scottish biologist, mathematician, and classics scholar well known for his varied interests and mildly eccentric habits. Nowadays he is probably best remembered for his pioneering work, On Growth and Form, published in 1917, in which he demonstrates that mathematical formulas and procedures can describe the forms of many living organisms and flowers. Mussels, for example, can be visualized as logarithmic spirals, and honey webs can be understood as the shapes with the smallest circumference from among all rectangles that cover an area without gaps.

But d’Arcy Thompson’s most astounding observation must have been that the shapes of quite different looking animals are often mathematically identical. Using the right coordinate transformation—that is, by tugging, pulling, or turning—a carp can be transformed into a moonfish. This also goes for other animals. The profiles of many four-legged creatures and birds differ from one another only because of the varying lengths and angles in their shapes.

D’Arcy Thompson’s explanation for this phenomenon was that different forces pull and squeeze the body until it assumes a streamlined figure or some other shape that is suitable for its environment. “Everything is the way it is because it got that way,” he wrote. The transformed shape is then passed on to the next generation. Since this can be interpreted as an adaptation to the environment, d’Arcy’s observations fit seamlessly into the already popular Darwinist worldview. (As an aside, the technique of emphasizing and distorting facial or bodily features—ears that stick out, oblong-shaped heads, big noses—has been

the caricaturist’s tool of the trade for centuries, to readers’ immeasurable delectation.)

The learned Scotsman was not the first to have used mathematics to describe natural phenomena. At the turn of the 13th century, Leonardo Bonacci from Pisa, later known as Fibonacci (son of Bonacci), had studied how rabbits multiply. The question he asked was how many pairs of rabbits coexist at any point in time, beginning with one pair of baby rabbits. At the end of the first month, the pair has reached puberty and the two rabbits mate. There is still only one pair of rabbits, but at end of the second month, the female gives birth to a new pair. There is now one adult pair and one baby pair. The adults mate again, thus producing another pair at the end of the third month. At this point there are three pairs of rabbits. One of these pairs has just been born, but the other two are old enough to mate, which, being rabbits, they do. A month later both they and their parents produce one additional pair each, and now a total of five pairs coexist. The answer to Bonacci’s question turns out to be a number sequence that from then on was called the Fibonacci series. Here are the first entries of the series: 1, 1, 2, 3, 5, 8, 13, 21, 34…. The process continues indefinitely, each number in the series being computed as the sum of the two predecessors (for example, 13 + 21 = 43). Of course, this does not prove that rabbits will take over the world, but only that Fibonacci had forgotten to factor in that rabbits tend to die after a certain time. (What this example does prove is that despite faultless induction mathematicians can sometimes abuse their science and arrive at erroneous conclusions simply by starting off with an incorrect premise.)

The Fibonacci series turns up in many guises. Kernels of sunflowers, for example, are arranged in left- and right-turning spirals. The number of kernels in the spirals usually corresponds to two consecutive numbers in the Fibonacci series—for example, 21 and 34. The number of spirals on pines and pineapples or the number of pricks on a cactus also corresponds to two consecutive numbers in the Fibonacci series. Nobody quite knows why this is

so, but there is a suspicion that the phenomenon is somehow connected to the efficacy of the plants’ growth.

Johan Gielis, a Belgian botanist who in the past was preoccupied with bamboo trees, decided to join the long line of scientists whose ambition it is to reduce natural phenomena to a single simple principle. His article, published in The American Journal of Botany, quickly caught the interest of the public, due in no small measure to the activation of a strong publicity machine and the catchy phrase—“the Superformula”—used in the article. In his work Gielis purports to show that many shapes found in living organisms can be reduced to a single geometrical form.

He starts off with an equation for a circle which, by adjusting a number of parameters, can be transformed into the equation for ellipses. Additional variations of the equation generate other shapes—triangles, squares, star shapes, concave and convex forms, and many other figures. Instead of tugging and tweaking at pictures as d’Arcy Thompson had done many years previously, Gielis tugged and tweaked at the six variables of his Superformula, thus simulating pictures of different plants and organisms. Because all the shapes that emerge do so through the transformations of the circle, they are, Gielis maintains, identical to each other.

The Superformula can by no means be defined as higher mathematics. Nor does it offer revolutionary insights or discoveries. Despite the media fuss and the accolades bestowed on it, the Superformula belongs more to hobby mathematics than to anything serious scientists could get excited about. At least Fibonacci had tried to give an explanation for his series in terms of the procreation of rabbits. And d’Arcy Thompson had come about his transformations by studying forces that supposedly act on an organism’s body. Gielis’s Superformula, on the other hand, offers no explanation whatsoever. It merely gives approximate descriptions of a number of organic forms. This drawback, however, did not keep the author from patenting the algorithm for his formula and creating a company to develop and market his invention.