Phyllotaxis is the study of the patterns on plants. The word itself comes from the Greek phullon, meaning "leaf," and taxis, meaning "arrangement." Phyllotaxis, in the restricted sense, is the study of the relative arrangement of what is called the primordia of plants. A primordium is, for example, what will become a leaf on a stem, a scale on a pinecone or on a pineapple fruit, a seed in the head (called the capitulum ) of a sunflower, or a floret in the capitulum of a daisy. In other words, phyllotaxis is the study of the patterns made by similar parts (such as florets, scales, and seeds) on plants and in their buds. Anatomically, phyllotactic patterns are closely related to the vascular systems of plants, but phyllotaxis-like patterns are even present in the brown alga Fucus spiralis, in which there is no vascular system. The study of phyllotaxis has brought about new ideas and considerable progress in our knowledge of the organization of vegetative shoots. Phyllotaxis was the oldest biological subject to be mathematized, well before genetics.
Types of Phyllotaxis
In the mid-1830s naturalists noticed the spirals in the capituli of daisies and sunflowers. There are indeed two easily recognizable families of spirals, winding in opposite directions with respect to a common pole that is the center of the capitulum. They also noticed the patterns of scales making families of spirals on the pineapple fruit surface. Depending on whether the scales are rectangular or hexagonal, there are two or three such families of spirals or helices that can be easily observed. These spirals are referred to as parastichies, meaning "secondary spirals." The accompanying figure of the Pinus pinea shows a cross-section of an apical bud with five parastichies in one direction and eight in the opposite direction. Similar patterns of helices are made by the points of insertions of the leaves around stems, such as the patterns of scars made by the leaves on the trunk of a palm tree.
Apart from the spiral or helical pattern, which is the type most frequently encountered in nature, there is another main type of phyllotaxis called whorled. A pattern is whorled when n primordia appear at each level of the stem, such as in horsetails (Equisetum ), in which n can take values from 6 to 20. When the n primordia on a level are inserted in between those of the adjacent level, the whorl is said to be alternating, as in fir club moss (Lycopodium selago ). When they are directly above those in the adjacent level, the whorl is called superposed, as in Ruta and Primula.
Numbers in Phyllotaxis
In the case of the spirals in the capitulum of the daisy, or in the case of those in the cross-section of the young pine cone in the figure, the spirals are often conceived as logarithmic spirals. In the case of mature pinecones and stems they are helices made by scales winding around a cylinder-like form. When naturalists count the spirals they find that in 92 percent of all the observations, the numbers of spirals are terms of the Fibonacci sequence, named after Leonardo Fibonacci, the most famous mathematician of the twelfth century. It is also called the main sequence. This is the recurrent sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, … where each term is the sum of the preceeding two. The next terms are thus 55 and 89, and the three dots (…) indicate that the sequence is infinite. Still more fascinating and puzzling is the fact that the number of spirals are consecutive terms of the Fibonacci sequence. For example, in the pine we have (2, 3), (5, 3), and (5, 8) phyllotaxes, in capituli the pairs found are (21, 34), (55, 34), (55, 89), and (89, 144), and on pineapples with hexagonal scales the triplets (8, 13, 21) or (13, 21, 34) are found, depending on the size of the specimens . The prevalence of the Fibonacci sequence in phyllotaxis is often referred to as "the mystery of phyllotaxis."
There are, of course, exceptions to the rule, but in the other cases of spiral (as opposed to whorled) phyllotaxis, the numbers obtained are consecutive terms of a Fibonacci-type sequence. This is a sequence of integers built on the same recurrence relationship as for the Fibonacci sequence, but starting with numbers different from 1 and 1, for instance: 1, 3, 4, 7, 11, 18, 29. … This sequence, encountered in Araucaria and Echinocactus, is present in about 1.5 percent of all observations, while the sequence 2, 2, 4, 6, 10, 16, 26 (the double Fibonacci sequence called the bijugate sequence) arises in around 6 percent of all the cases and is observed for example in Aspidium and Bellis. The phenomenon of phyllotaxis is thus essentially simple as far as those sequences are concerned, but the matter becomes complicated when on the same plant, such as Bryophyllum and Anthurium, one observes many Fibonacci-type sequences. This phenomenon is referred to as discontinuous transition. In the capituli of sunflowers and daisies, transitions are made along the same sequence. For example, we can observe in the center of the head (5, 8) phyllotaxis, followed in the middle part by (13, 8) phyllotaxis, and in the outer part by (13, 21) phyllotaxis. This is called a continuous transition. This phenomenon of growth has to do with the way crystals grow, and the daisy can be considered a living crystal.
Stems of Leaves and the Golden Number.
Let us consider now a stem of leaves, as naturalists did in the mid-1830s. Take a point of insertion of a leaf at the bottom of the stem, and, in a helical or spiral movement around the stem, go to the next leaves above by the shortest path from one leaf to the next until a leaf is reached that is directly above the first chosen one. The leaves are then linked consecutively (1, 2, 3, 4, 5, …) along a helix, while in the case of the pine cone in the figure the five parastichies link the primordia by steps of five (e.g., 0, 5, 10, 15, 20, …). Then by making the ratio of the number of turns around the stem to the number of leaves met, excluding the first one, we obtain a fraction, such as 2/5, illustrated in the accompanying figure of the stem. In a significant number of cases the fractions obtained on various stems are 1/2, 1/3, 2/5, 3/8, 5/13, 8/21. … The numerators and the denominators of this sequence of fractions are consecutive terms of the Fibonacci sequence. Each fraction represents an angle d between two consecutive leaves along the helix, known as the divergence angle. In the case of the pine cone the divergence is the angle between consecutively numbered primordia such as #24 and #25, and using a protractor it can be checked that d ≅ 137.5 degrees, which is known as the Fibonacci angle.
These divergences are closely related to what is known as the golden number, denoted by the Greek letter Τ (tau), where Τ ≅ 1.618. Indeed, the value of 1/Τ2 ≅ 0.382, which is the value the sequence of fractions approaches. For example, 5/13 ≅ 0.384 or 8/21 ≅ 0.380, and as we take fractions farther away in the sequence, such as 21/55, we find that 21/55 ≅ 0.381 and that we are gradually approaching the value of 1/Τ2. Also the value of 360/Τ2 ≅ 137.5.
Phyllotaxis and Explanatory Modeling.
The aim of explanatory modeling is to try to reproduce the patterns from rules or mechanisms or principles— imagined or hypothesized by the modeler—that are considered to be in action in shoot apices. The hypotheses are then transcribed into mathematical terms and their consequences are logically drawn and compared to reality. Two old hypotheses in particular have been scrutinized in different manners by the modelers. One is the chemical hypothesis that a substance such as a plant hormone produced by the primordia and the tip of the apex is at work, inhibiting the formation of primordia at some places and promoting their formation at others, thus producing the patterns. Another stresses the idea that physical-contact pressures between the primordia generate the patterns. A new hypothesis suggests that elementary rules of growth such as branching, and elementary principles such as maximization of energy, are at work producing the patterns. This model predicts the existence of a very unusual type of pattern, known as monostichy, in which all the primordia would be superimposed on the same side of the stem. This type of pattern was later discovered to exist in Utricularia. The same model shows the unity behind the great diversity of patterns.
Phyllotaxis is clearly a subject at the junction of botany and mathematics. Mathematics helps to organize the data, give meaning to it, interpret it, and direct attention to potentially new observations. The study of phyllotaxis has become a multidisciplinary subject, involving general comparative morphology, paleobotany, genetics, molecular biology, physics, biochemistry, the theory of evolution, dynamical system theory , and even crystallography . The patterns observed in plants can be seen to a much lesser extent in other areas of nature.
see also Anatomy of Plants; Leaves; Shape and Form of Plants; Stems.
Roger V. Jean
Church, A. H. On the Relation of Phyllotaxis to Mechanical Laws. London: Williams and Norgate, 1904.
Jean, Roger V. Phyllotaxis: A Systemic Study in Plant Morphogenesis. Cambridge: Cambridge University Press, 1994.
——, and Barabé Denis, eds. Symmetry in Plants. Singapore: World Scientific Publishing, 1998.