## ꩜ A Brief History of Phyllotaxis ꩜

꩜ Ancient Egyptians were the source of Greek science, and as skilled observers, probably knew about numbers and patterns in plants and the number *t*.

**370-285 BC**Theophrastus wrote

*Enquiry into Plants*which mentions leaves in regular series.

**23-79 AD**Pliny wrote

*Natural History*which includes more detailed descriptions of plant patterns as a way of categorizing plants.

꩜ **1202** Leonardo Fibonacci obtained the Fibonacci sequence <1, 1, 2, 3, 5, … *F*(*k*), *F*(*k*+1)> as a solution to the problem of monthly population growth of rabbits. This sequence relates to t by the formula lim [*F*(*k*+1)/*F*(*k*)] = *t**.*

* *

꩜ **1452-1519** Leonardo Da Vinci anticipated, in one of his notebooks, the observation of spiral patterns in plants such as those made by Bonnet.

꩜ **1571-1630** Johannes Kepler was fascinated by the number 5 and incorrectly concluded that the Fibonacci sequence propagated itself as the seeding capacity of plants, but did correctly surmise the intrinsic involvement of the sequence in plant growth.

꩜ **1754** Charles Bonnet in his *Recherches sur l’Usage des Feuilles dans les Plantes *mentions the different phyllotactic arrangements of leaves, the genetic spiral, and one family of parastichies. He initiated observational phyllotaxis.

꩜ **1830** Schimper introduced the concepts of the genetic spiral, the divergence angle, and the parastichy. He defined the divergence angle as a fraction by dividing the spiral arrangement into cycles and setting *d* = times around the stem per cycle/leaves per cycle.

꩜ **1831, 1835** Alexander Braun observed conspicuous parastichies on pine cones, noting that the number of left and right parastichies were generally consecutive Fibonacci numbers.

꩜ **1837** Auguste and Louis Bravais used a point lattice on a cylinder to represent leaf distribution, found the number of genetic spirals to be equal to the greatest common divisor of the numbers of secondary spirals, and defined the divergence angle as an irrational number related to the sequence from which the numbers of secondary spirals came.

꩜ **1848** Lestiboudois found a connection between phyllotactic patterns and the phenomena of branching and ramification.

꩜ **1868** Hofmeister proposed that new primordia form periodically at the apex boundary in the largest gap left by the preceding primordia.

꩜ **1872** P. G. Tait rewrote and interpreted the Bravais brothers’ results concerning the divergence angle and parastichy pair.

꩜ **1873** Airy explained the arrangement of leaf primordia in a bud in terms of economy of space.

꩜ **1875** Weisner theorized that the divergence between leaves evolved by natural selection as a way to maximize leaf exposure to light.

꩜ **1878** Simon Schwendener described conspicuous opposed parastichy pairs, observed the changes in divergence angle and the ratio of internode distance to stem girth as the plant growth leads to higher phyllotaxes, and explained (incorrectly) that leaf arrangements are the result of contact pressure between primordia.

꩜ **1882** Julius Sachs rejected the mathematical theory of phyllotaxis, seeing no significance in the continuous fractions for divergence angles.

꩜ **1904** Church proposed an explanation for the phenomena of transitions between phyllotactic patterns, and that parastichies are lines of force. He rejected the genetic spiral and the cylinder model in favor of parastichies and a centric representation.

꩜ **1907** Van Iterson constructed a model of leaf primordia around a cylinder which assumed close packing.

꩜ **1913** Schoute concluded that the dominance of the Fibonacci sequence over parastichy numbers was still unexplained and that Schwendener’s work headed to the solution. He explained the placement of new primordia as a reaction to chemical inhibitors released by primordia to keep others from growing too close.

꩜ **1917** D’Arcy Thompson proclaimed that there was no reason to prefer any one parastichy or family of parastichies and transformed the whole subject into speculation by concluding that there was an irreducible subjectivity to it.

꩜ **1931** Mary and Robert Snow concluded from studying the effects of isolating leaf primordia that the positioning of new primordia is affected by adjacent, preexisting primordia.

꩜ **1948, 1951** Richards introduced the idea of the plastochrone ratio and developed a system of equations to mathematically describe a centric representation using three parameters: plastochrone ratio, divergence angle, and the angle of the cone tangential to the apex in the area being considered.

꩜ **1950** Plantefol concentrated on one family of spirals (foliar helices) and their biological reality without any mathematics.

꩜ **1974** Adler wrote out the Fundamental Theorem of Phyllotaxis.

꩜ **1983** Erickson rediscovered and extended Van Iterson’s analysis.

꩜ **1984** Roger Jean reworked the Fundamental Theorem of Phyllotaxis.

꩜ **1991** Levitov found and explained phyllotactic patterns in a flux lattice of a superconductor as a way of minimizing energy in the system.

꩜ **1992** Douady and Couder found that drops of ferrofluid placed periodically in the center of a dish produced phyllotactic patterns.