Editing Phyllotaxis (section)

==Mathematics==
[[File:phyllotaxis_golden_angle.svg|thumb|End-on view of a plant stem showing consecutive leaves separated by the [[golden angle]] ]]
Physical models of phyllotaxis date back to [[George Biddell Airy|Airy]]'s experiment of packing hard spheres. [[Gerrit van Iterson]] diagrammed grids imagined on a cylinder (rhombic lattices).<ref>{{cite web | url=http://www.math.smith.edu/phyllo/About/History/VanIterson01.html | title=History | publisher=Smith College | access-date=24 September 2013 | archive-url=https://web.archive.org/web/20130927224034/http://www.math.smith.edu/phyllo/About/History/VanIterson01.html | archive-date=27 September 2013 | url-status=dead }}</ref> Douady et al. showed that phyllotactic patterns emerge as self-organizing processes in dynamic systems.<ref>{{cite journal | vauthors = Douady S, Couder Y | title = Phyllotaxis as a physical self-organized growth process | journal = Physical Review Letters | volume = 68 | issue = 13 | pages = 2098–2101 | date = March 1992 | pmid = 10045303 | doi = 10.1103/PhysRevLett.68.2098 | bibcode = 1992PhRvL..68.2098D }}</ref> In 1991, Levitov proposed that lowest energy configurations of repulsive particles in cylindrical geometries reproduce the spirals of botanical phyllotaxis.<ref>{{cite journal |author=Levitov LS |title=Energetic Approach to Phyllotaxis |journal=Europhys. Lett. |volume=14 |issue=6 |pages=533–9 |date=15 March 1991 |doi= 10.1209/0295-5075/14/6/006 |bibcode = 1991EL.....14..533L |s2cid=250864634 }}<br/>
{{cite journal | vauthors = Levitov LS | title = Phyllotaxis of flux lattices in layered superconductors | journal = Physical Review Letters | volume = 66 | issue = 2 | pages = 224–227 | date = January 1991 | pmid = 10043542 | doi = 10.1103/PhysRevLett.66.224 | bibcode = 1991PhRvL..66..224L }}</ref> 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".<ref>{{cite journal | vauthors = Nisoli C, Gabor NM, Lammert PE, Maynard JD, Crespi VH | title = Static and dynamical phyllotaxis in a magnetic cactus | journal = Physical Review Letters | volume = 102 | issue = 18 | pages = 186103 | date = May 2009 | pmid = 19518890 | doi = 10.1103/PhysRevLett.102.186103 | arxiv = cond-mat/0702335 | bibcode = 2009PhRvL.102r6103N | s2cid = 4596630 }}</ref><ref>{{cite journal | vauthors = Nisoli C | title = Spiraling solitons: A continuum model for dynamical phyllotaxis of physical systems | journal = Physical Review E | volume = 80 | issue = 2 Pt 2 | pages = 026110 | date = August 2009 | pmid = 19792203 | doi = 10.1103/PhysRevE.80.026110 | url = https://www.researchgate.net/publication/26858941 | arxiv = 0907.2576 | bibcode = 2009PhRvE..80b6110N | s2cid = 27552596 }}</ref> 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 [[roton]]s and [[Maxon excitation|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.<ref name="Matila Ghyka 1946">{{Cite book | title = The Geometry of Art and Life | first = Matila | last = Ghyka | name-list-style = vanc | publisher = Dover | isbn = 978-0-486-23542-4 | url = https://archive.org/details/geometryofartlif00mati | year = 1977 }}</ref><ref>{{cite book|last=Adler|first=Irving|name-list-style=vanc|title=Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur On Plants|title-link= Solving the Riddle of Phyllotaxis}}</ref>