Inbotany,phyllotaxis (from Ancient Greek φύλλον (phúllon) 'leaf' and τάξις (táxis) 'arrangement')[1] orphyllotaxy is the arrangement ofleaves on aplant stem. Phyllotactic spirals form a distinctive class ofpatterns in nature.
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The basicarrangements of leaves on a stem areopposite andalternate (also known asspiral). Leaves may also bewhorled if several leaves arise, or appear to arise, from the same level (at the samenode) on astem. With an opposite leaf arrangement, two leaves arise from the stem at the same level (at the samenode), on opposite sides of the stem. An opposite leaf pair can be thought of as a whorl of two leaves. With an alternate (spiral) pattern, each leaf arises at a different point (node) on the stem.
Distichous phyllotaxis, also called "two-ranked leaf arrangement", resembles a fan-shape in form and is a special case of either opposite or alternate leaf arrangement where the leaves on a stem are arranged in two vertical columns on opposite sides of the stem. Examples include theStrelitziaceae, where this leaf arrangement is a feature.[2]
In an opposite pattern, if successive leaf pairs are 90 degrees apart, this habit is calleddecussate. It is common in members of the familyCrassulaceae[3] Decussate phyllotaxis also occurs in theAizoaceae. In genera of the Aizoaceae, such asLithops andConophytum, many species have just two fully developed leaves at a time, the older pair folding back and dying off to make room for the decussately oriented new pair as the plant grows.[4] If the arrangement is both distichous and decussate, it is calledsecondarily distichous.
The whorled arrangement is fairly unusual on plants except for those with particularly shortinternodes. Examples of trees with whorled phyllotaxis areBrabejum stellatifolium[5] and the related genusMacadamia.[6] A whorl can occur as abasal 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 arosette.
The rotational angle from leaf to leaf in a repeating spiral can be represented by a fraction of afull rotation around the stem.
Alternate distichous leaves will have an angle of 1/2 of a full rotation. Inbeech and verticalhazel twigs the angle is 1/3, inoak andapricot it is 2/5, insunflowers,poplar, andpear, it is 3/8, and inwillow andalmond the angle is 5/13.[7] The numerator and denominator normally consist of aFibonacci number and its second successor. 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 pattern becomes complex and non-repeating. This tends to occur with a basal configuration. Examples can be found incompositeflowers andseed heads. The most famous example is thesunflower head. This phyllotactic pattern creates an optical effect 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 beFibonacci numbers. In some cases, the numbers appear to be multiples of Fibonacci numbers because the spirals consist of whorls.
Some early scientists—notablyLeonardo da Vinci—made observations of the spiral arrangements of plants.[8] In 1754,Charles Bonnet observed that the spiral phyllotaxis of plants were frequently expressed in bothclockwise and counter-clockwisegolden ratio series.[9] Mathematical observations of phyllotaxis followed withKarl Friedrich Schimper and his friendAlexander Braun's 1830 and 1830 work, respectively;Auguste Bravais and his brother Louis connected phyllotaxis ratios to theFibonacci sequence in 1837.[9]
Insight into the mechanism had to wait untilWilhelm Hofmeister proposed a model in 1868. Aprimordium, the nascent leaf, forms at the least crowded part of the shootmeristem. Thegolden angle between successive leaves is the blind result of this jostling. Since three golden arcs add up to slightly more than enough to wrap a circle, this guarantees that no two leaves ever follow the same radial line from center to edge. The generative spiral is a consequence of the same process that produces the clockwise and counter-clockwise spirals that emerge in densely packed plant structures, such asProtea flower disks or pinecone scales.
In the early 20th century, researchers such asMary Snow and George Snow continued these lines of inquiry.[10]
The pattern of leaves on a plant is ultimately controlled by the accumulation of the plant hormoneauxin in certain areas of themeristem.[11][12] Leaves become initiated in localized areas where auxin concentration is higher.[disputed –discuss] When a leaf is initiated and begins development, auxin begins to flow towards it, thus depleting auxin from area on themeristem close to where the leaf was initiated. This gives rise to a self-propagating system that is ultimately controlled by the ebb and flow of auxin in different regions of themeristematictopography.[13][14]
Physical models of phyllotaxis date back toAiry's experiment of packing hard spheres.Gerrit van Iterson diagrammed grids imagined on a cylinder (rhombic lattices).[15] Douady et al. showed that phyllotactic patterns emerge as self-organizing processes in dynamic systems.[16] In 1991, Levitov proposed that lowest energy configurations of repulsive particles in cylindrical geometries reproduce the spirals of botanical phyllotaxis.[17] More recently, Nisoli et al. (2009) showed that to be true by constructing a "magnetic cactus" made of magnetic dipoles mounted on bearings stacked along a "stem".[18] They demonstrated that these interacting particles can access novel dynamical phenomena beyond what botany yields: a "dynamical phyllotaxis" family of non local topologicalsolitons emerge in thenonlinear regime of these systems, as well as purely classicalrotons andmaxons in the spectrum of linear excitations.[18][19]
Close packing of spheres generates a dodecahedral tessellation with pentaprismic faces. Pentaprismic symmetry is related to the Fibonacci series and thegolden section of classical geometry.[20][21]
Phyllotaxis has been used as an inspiration for a number of sculptures and architectural designs. Akio Hizume has built and exhibited several bamboo towers based on the Fibonacci sequence which exhibit phyllotaxis.[22] Saleh Masoumi has proposed a design for an apartment building in which the apartmentbalconies project in a spiral arrangement around a central axis and none shade the balcony of the apartment directly beneath.[23]
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