84
Chamberlain, in The Living Cycads, described how he estimated the age of a Dioon edule, which reaches maturity (in the wild) around the age of fifty, and then puts out a new crown of leaves every other year on average. By counting the number of leaf scales on the stem, and dividing by the number of leaves produced each year, he arrived at the age of the tree. He described one beautiful specimen which, by this criterion, was 970 years old, even though less than five feet in height. Indeed, Chamberlain wondered whether some cycads might approach the sequoias in age.
85
The cones of cycads vary in character and shape and size: the vast cones of Lepidozamia peroffskyana and Encephalartos transvenosus may weigh more than a hundred pounds, and the cones of the smallest Za-mias no more than thirty milligrams. But all of them exhibit, in the arrangement of their cone scales, intricate geometric patterns similar to the corkscrew spirals or helices we see in pinecones, the leaf arrangement of cylindrical stems, or the whorling florets of sunflowers. The study of these patterns, this phyllotaxis, has intrigued botanists and mathematicians for centuries, not only because the spirals themselves are logarithmic, but because there are numbers of accessory helices (or parastichies) running in the opposite direction and these two sets of helices occur in a fixed ratio to one another. Thus in cycad cones, as in pinecones, we almost always see spirals in five and eight rows, and if we express as fractions the number of parastichies, we find a series of 2⁄1, 3⁄2, 5⁄3, 8⁄5, 13⁄8, 21⁄13, 34⁄21, and so on. This series, named after the thirteenth-century mathematician Fibonacci, corresponds to a continued fraction which converges to 1.618, the numerical equivalent of the Golden Section.
These patterns probably represent no more and no less than an optimum way of packing leaves or scales together while avoiding their superimposition (and not, as Goethe and others thought, some mystical archetype or ideal), but they are a delight to the eye and a stimulus to the mind. Phyllotaxis fascinated the Reverend J.S. Henslow (professor of botany at Cambridge, and Darwin’s teacher), who discussed and illustrated it in his Principles of Descriptive and Physiological Botany, and it is pondered at length in an eccentric (and very favorite) book, D’Arcy Thompson’s On Growth and Form. It is said that Napier’s discovery of logarithms at the start of the seventeenth century was stimulated by a contemplation of the growth of horsetails, and the great botanist Nehemiah Grew, later in the century, observed that ‘from the contemplation of Plants, men might first be invited to Mathematical Enquiry.’
This sense of the mathematical determination (or constraints) of nature, especially of organic form and growth, divested of idealism or idiosyncrasy, is very strong now, especially with the development of chaos and complexity theory in the last few decades. Now that fractals are, so to speak, part of our consciousness, we see them everywhere – in mountains, in landscapes, in snowflakes, in migraines, but above all in the vegetable world – just as Napier, four centuries ago, saw logarithms in his garden, and Fibonacci, seven centuries ago, found the Golden Section all about him.
86
The forms of plants exercised Goethe endlessly – we owe the very word ‘morphology’ to him. He had no sense of evolution, but rather of a sort of logical or morphological calculus whereby all higher plants might be derived from a simple primordial type, a hypothetical ancestral plant he called an Ur-pflanze. (This idea came to him, he recorded, while he was gazing at a palm in the Orto at Padua, and ‘Goethe’s palm,’ as it is now called, still grows there in a house of its own.) His hypothetical Ur-pflanze had leaves, which could metamorphose into petals and sepals, stamens, and anthers, all the complex parts of flowers. Had Goethe concerned himself with flowerless plants, I could not help feeling, he might have seized on Psilotum as his Ur-pflanze.
Alexander von Humboldt was a close friend of Goethe’s, and adopted his theory of metamorphosis in his own Physiognomy of Plants (indeed, he widens Goethe’s notion and hints at a cosmic, universal organizing power acting not only on plants but on the forms of rocks and minerals and on the forms of mountains and other natural features as well). The physiognomy of the vegetable kingdom, he argues, ‘is principally determined by sixteen forms of plants.’ One of these – a leafless branching form – to his mind, binds together plants as diverse as the Casuarinas (flowering plants), Ephedra (a primitive gymnosperm) and Equisetum (a horsetail). Humboldt was a superb practical botanist, and very well appreciated the botanical differences between these, but he was looking, as Goethe was, for a principle orthogonal to biology, to all particular sciences – a general principle of morphogenesis or morphological constraints.
The arborization of plants originates not in accordance with some primordial archetype, but as the simplest geometric way of maximizing the ratio of surface area to volume and thus the area available for photosynthesis. Similar economic considerations may apply to many biological forms, such as the branching dendrites of nerve cells or the arborizations of the respiratory ‘tree.’ Thus an ‘Ur’ – plant like Psilotum, lacking leaves or other complications, is an exemplar, a diagram of one of nature’s most basic structures.
(In more recent times, a specific analog of Goethe’s theory, which traces how all higher plants might be derived morphologically from primitive psilophytes, has been proposed by W. Zimmerman, in his theory of telomes. And a general analog to Goethe’s morphology may be found in some of the current theories of self-organization, complexity, and universal morphogenesis.)
87
Such a feeling of transport to the distant past struck Safford when he saw the cycad forests of Guam: their ‘cylindrical, scarred trunks, and stiff, pinnated, glossy leaves,’ he wrote, suggested ‘ideal pictures of the forests of the Carboniferous age.’
A very similar feeling is described by John Mickel, writing of horsetails:
To wander among them is a kind of science-fiction experience. I well remember the first time I encountered a stand of the giant horsetail in Mexico. I had the feeling that I had found my way backward into a Carboniferous forest, and half expected dinosaurs to appear among the horsetails.
Even a walk in the streets of New York can evoke the Paleozoic: one of the commonest trees here (apparently well able to resist pollution) is the maidenhair tree, Ginkgo biloba, a unique survivor little changed from the ginkgophytes of the Permian. But the ginkgo exists now only in cultivation; it is no longer found in the wild.