Phyllotaxis
In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon "leaf" and táxis "arrangement").^{[1]} Phyllotactic spirals form a distinctive class of patterns in nature.
Contents
Leaf arrangement
The basic arrangements of leaves on a stem are opposite and alternate (also known as 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.
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 "tworanked 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. It also occurs in other plant habits such as those of Gasteria or Aloe seedlings, and also in mature plants of related species such as Kumara plicatilis (Aloe plicatilis).
In an opposite pattern, if successive leaf pairs are 90 degrees apart, this habit is called decussate. It is common in members of the family Crassulaceae^{[2]} Decussate phyllotaxis also occurs in the Aizoaceae. In genera of the Aizoaceae, such as Lithops and Conophytum, 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.^{[3]}
The whorled arrangement is fairly unusual on plants except for those with particularly short internodes. Examples of trees with whorled phyllotaxis are Brabejum stellatifolium^{[4]} and the related genus Macadamia.^{[5]}
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.^{[6]} 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 nonrepeating. 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 crisscrossing spirals. In the botanical literature, these designs are described by the number of counterclockwise 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.
Determination
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.^{[7]} Leaves become initiated in localized areas where auxin is absent.^{[disputed – discuss]} 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 selfpropagating system that is ultimately controlled by the ebb and flow of auxin in different regions of the meristematic topography.^{[8]}
History
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 counterclockwise 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^{[9]} 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^{[citation needed]} and occasionally the leaf positioning appears to be random.
Mathematics
Physical models of phyllotaxis date back to Airy's experiment of packing hard spheres. Gerrit van Iterson diagrammed grids imagined on a cylinder (Rhombic Lattices).^{[10]} Douady et al. showed that phyllotactic patterns emerge as selforganizing processes in dynamic systems.^{[11]} In 1991, Levitov proposed that lowest energy configurations of repulsive particles in cylindrical geometries reproduce the spirals of botanical phyllotaxis.^{[12]} 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".^{[13]}^{[14]} They demonstrated that these interacting particles can access novel dynamical phenomena beyond what botany yields: a "Dynamical Phyllotaxis" family of non local topological solitons emerge in the nonlinear regime of these systems, as well as purely classical rotons and maxons in the spectrum of linear excitations.
Close packing of spheres generates a dodecahedral tessellation with pentaprismic faces. Pentaprismic symmetry is related to the Fibonacci series and the golden section of classical geometry.^{[15]}^{[16]}
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.^{[17]} 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.^{[18]}
See also
Wikimedia Commons has media related to phyllotaxis. 
Notes
 ^ φύλλον, τάξις. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project
 ^ Urs Eggli (6 December 2012). Illustrated Handbook of Succulent Plants: Crassulaceae. Springer Science & Business Media. pp. 40–. ISBN 9783642558740.
 ^ Heidrun E.K. Hartmann (6 December 2012). Illustrated Handbook of Succulent Plants: Aizoaceae A–E. Springer Science & Business Media. pp. 14–. ISBN 9783642563065.
 ^ Marloth, Rudolf. The Flora of South Africa” 1932 Pub. Capetown: Darter Bros. London: Wheldon & Wesley.
 ^ Chittenden, Fred J. Ed., Royal Horticultural Society Dictionary of Gardening, Oxford 1951
 ^ Coxeter, H. S. M. (1961). Introduction to geometry. Wiley. p. 169.
 ^ Traas J, Vernoux T (June 2002). "The shoot apical meristem: the dynamics of a stable structure". Philosophical Transactions of the Royal Society B. 357 (1422): 737–47. doi:10.1098/rstb.2002.1091. PMC 1692983 . PMID 12079669.
 ^ Smith, RS (2008). "The role of Auxin Transport in Plant Patterning Mechanisms". PLoS Biol. 6 (12): e323. doi:10.1371/journal.pbio.0060323. PMC 2602727 . PMID 19090623.
 ^ Snow, M.; Snow, R. (1934). "The interpretation of Phyllotaxis". Biological Reviews. 9 (1): 132–137. doi:10.1111/j.1469185X.1934.tb00876.x.
 ^ "History". Smith College. Retrieved 24 September 2013.
 ^ Douady S, Couder Y (March 1992). "Phyllotaxis as a physical selforganized growth process". Phys. Rev. Lett. 68 (13): 2098–2101. Bibcode:1992PhRvL..68.2098D. doi:10.1103/PhysRevLett.68.2098. PMID 10045303.

^ Levitov LS (15 March 1991). "Energetic Approach to Phyllotaxis" (PDF). Europhys. Lett. 14 (6): 533–9. Bibcode:1991EL.....14..533L. doi:10.1209/02955075/14/6/006.
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:condmat/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 . Bibcode:2009PhRvE..80b6110N. doi:10.1103/PhysRevE.80.026110.
 ^ Matila Ghyka. The Geometry of Art and Life. Dover. ISBN 0486235424.
 ^ Adler, Irving. Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur On Plants. Retrieved 8 June 2012.
 ^ Akio Hizume. "Star Cage". Retrieved 18 November 2012.
 ^ "Open to the elements". World Architecture News.com. 11 Dec 2012.
References
 F.M.J. van der Linden
 van der Linden FMJ (1 April 1996). "Creating phyllotaxis: The stackanddrag model". Mathematical Biosciences. 133 (1): 21–50. doi:10.1016/00255564(95)000771. PMID 8868571.
 Frank M.J. van der Linden (1998). "Creating Phyllotaxis from Seed to Flower". In Denis Barabe; Jean, Roger V. Symmetry in Plants. World Scientific Series in Mathematical Biology and Medicine. 4. Singapore: World Scientific Pub Co Inc. ISBN 9810226217.
External links
 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 Lsystem using JSXgraph
 Phyllotaxis: An Interactive Site for the Study of Plant Pattern Formation
 Interactive Parastichies Explorer to build phyllotactic spirals
 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