In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon "leaf" and táxis "arrangement"). Phyllotactic spirals form a distinctive class of patterns in nature.
Pattern structure 
The basic phyllotactic patterns are opposite, or alternate = spiral. Leaves may also be whorled if several leaves arise, or appear to arise, from the same level (at the same node) on a stem. This arrangement is fairly unusual on plants except for those with particularly short internodes. Examples of a trees with whorled phyllotaxis are Brabejum stellatifolium and the related Macadamia genus.
With an opposite leaf arrangement, two leaves arise from the stem at the same level (at the same node), 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" 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 various bulbous plants such as Boophone, Aloe seedlings, and also mature Aloe plicatilis.
In an opposite pattern, if successive leaf pairs are perpendicular, this is called decussate.
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.
Repeating spiral 
A repeating spiral can be represented by a fraction describing the angle of windings leaf per leaf.
Alternate distichous leaves will have an angle of 1/2 of a full rotation. In beech and hazel the angle is 1/3, in oak and apricot it is 2/5, in sunflowers, poplar, and pear, it is 3/8, and in willow and almond the angle is 5/13. The numerator and denominator normally consist of a Fibonacci 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 in composite flowers and seed heads. The most famous example is the sunflower 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 be Fibonacci numbers. In some cases, the numbers appear to be multiples of Fibonacci numbers because the spirals consist of whorls.
The pattern of leaves on a plant is ultimately controlled by the local depletion of the plant hormone auxin in certain areas of the meristem. Leaves become initiated in localized areas where auxin is absent. When a leaf is initiated and begins development, auxin begins to flow towards it, thus depleting auxin from another area on the meristem where a new leaf is to be initiated. This gives rise to a self-propagating system that is ultimately controlled by the ebb and flow of auxin in different regions of the meristematic topography.
Insight into the mechanism had to wait until Wilhelm Hofmeister proposed a model in 1868. A primordium, the nascent leaf, forms at the least crowded part of the shoot meristem. The golden 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 as Protea flower disks or pinecone scales.
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.
Phyllotaxis and physics 
Physical models of phyllotaxis date back to Airy's experiment of packing hard spheres. Douady et al. also have shown experimentally and theoretically that phyllotactic patterns emerge as self-organizing processes in dynamic systems. In 1991, Levitov proposed that lowest energy configurations of repulsive particles in cylindrical geometries reproduce the spirals of botanical phyllotaxis. More recently, Nisoli et al. (2009) have shown experimentally and numerically that indeed that was the case, by constructing a "magnetic cactus" made of magnetic dipoles mounted on bearings stacked along a "stem". They also revealed that these interacting particles can access novel dynamical phenomena beyond what botany yields: a family of highly non local novel topological solitons emerge in the nonlinear regime of these systems, as well as purely classical rotons and maxons in the spectrum of linear excitations. They named these novel phenomena "Dynamical Phyllotaxis", as they appear in physical systems whose statics is dictated by the number theoretical laws of phyllotaxis. Close packing of spheres also generates a dodecahedral tesselation with pentaprismic faces. Pentaprismic symmetry is intimately related to the Fibonacci series and the golden section of classical geometry. This relationship has been explored at length by Matila Ghyka.
Phyllotaxis and Mathematics 
The term Phyllotaxis refers to the patterns on plants formed by the arrangement of repeated biological units. In nearly all cases, the Fibonacci Numbers and the Golden Ratio occur in these arrangements. Over a period of more than two decades, Irving Adler wrote papers that construct a rigorously derived mathematical model for Phyllotaxis, which are major and enduring contributions to the field.
Phyllotaxis in art and architecture 
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. Saleh Masoumi has proposed a design for an apartment building where the apartment balconies project in a spiral arrangement around a central axis and each one does not shade the balcony of the apartment directly beneath.
See also 
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Levitov LS (January 1991). "Phyllotaxis of flux lattices in layered superconductors". Phys. Rev. Lett. 66 (2): 224–7. Bibcode:1991PhRvL..66..224L. doi:10.1103/PhysRevLett.66.224. PMID 10043542.
- Nisoli C, Gabor NM, Lammert PE, Maynard JD, Crespi VH (May 2009). "Static and dynamical phyllotaxis in a magnetic cactus". Phys. Rev. Lett. 102 (18): 186103. arXiv:cond-mat/0702335. Bibcode:2009PhRvL.102r6103N. doi:10.1103/PhysRevLett.102.186103. PMID 19518890.
- Nisoli C, (August 2009). "Spiraling solitons: A continuum model for dynamical phyllotaxis of physical systems". Phys. Rev. E 80 (2): 026110. arXiv:0907.2576. doi:10.1103/PhysRevE.80.026110.
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- Fibonacci Numbers and the Golden Section
- Phyllotaxis as a Dynamical Self Organizing Process
- Weisstein, Eric W., "Phyllotaxis", MathWorld.
- Phyllotaxis Spirals and Phyllotaxis Spirals in 3D by Stephen Wolfram, The Wolfram Demonstrations Project.
- An interactive L-system using JSXgraph
- Phyllotaxis: An Interactive Site for the Study of Plant Pattern Formation
- Magnetic Cactus Experimentally Demonstrates Mathematical Plant Patterns
- Links between Phyllotaxis and the Prime numbers
- Solving the Riddle of Phyllotaxis - Why The Fibonacci Numbers And The Golden Ratio Occur On Plants