Phyllotaxis
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In botany, phyllotaxis or phyllotaxy is the arrangement of the leaves on the shoot of a plant. The basic patterns are alternate, opposite, whorled or spiral. With an alternate pattern, leaves switch from side to side. A alternate distichous phyllotaxis means that each leave growing at a single node is disposed in a single rank along the branch (such as in grasses). In an opposite pattern, two leaves grow in opposite directions from the same node. In an opposite pattern, if successive leaf pairs are perpendicular, this is called decussate. A whorled pattern consists of three or more leaves at each node. An opposite leaf pair can be thought of as a whorl of two leaves. A whorl can occur as a basal structure where all the leaves are attached at the base of the shoot and the internodes are small or nonexistent. A basal whorl with a large number of leaves spread out in a circle is called a rosette. A multijugate pattern is a spriral composed of whorls.
A repeating spiral can be represented by a fraction describing the angle of windings leaf per leaf. The numerator and denominator normally consist of a Fibonacci number and its second successor. Examples are 3/5 and 5/8. The number of leaves is sometimes called rank, in the case of simple Fibonacci ratios, because the leaves line up in vertical rows. With larger Fibonacci pairs, the ratio approaches phi and the pattern becomes complex and non-repeating. This tends to occurs with a basal configuration. Examples can be found in composite flowers and seed heads. The most famous example is the sunflower head. This phyllotactic pattern creates an optical illusion of criss-crossing spirals. In the botanical literature, these designs are described by the number of counter-clockwise spirals and the number of clockwise spirals. These also turn out to be Fibonacci numbers. In some cases, the numbers appear to be multiples of Fibonacci numbers because the spirals consist of whorls.
Leonardo da Vinci was the first to suggest that the adaptive advantage of the Fibonacci pattern is to maximize exposure to dew. Current thinking supports this interpretation. Phyllotactic architecture optimizes access to moisture, rainfall and sunlight.
Insight into the mechanism had to wait until Hoffmeister proprosed a model in 1868. The process begins with two primordia, nascent leaves, forming on opposite sides of the shoot. A third new leaf then erupts between them. Each old leaf pushes the upstart away. The golden angle is the blind result of this jostling. Since three golden arcs add up to slightly more than enough to wrap a circle, one of the old leaves overlaps and is pushed out in a radial line, like a rocket that escapes earth orbit. This vacates space in the inner generative spiral for a new leaf to form. The generative spiral should not be confused with the clockwise and counter-clockwise spirals that emerge in densely packed plant structures. Spirals are discerned by tracing the path from leaf to neighboring leaf and each leaf acquires new neighbors as it shoots out from the center and its sister primordia. New leaves move into the spaces opened as old leaves diverge. Phi is an irrational number and this guarantees that no two leaves ever follow the same radial line from center to edge.
In modern times, researchers such as Snow and Snow have continued these lines of inquiry. Computer modeling and morphological studies have confirmed and refined Hoffmeister's ideas. Questions remain about the details. Botanists are divided on whether the control of leaf migration depends on chemical gradients among the primordia or purely mechanical forces. Lucas rather than Fibonacci numbers have been observed in a few plants and occasionally the leaf positioning appears to be random.
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Fibonacci Numbers and the Golden Section
