A Brief History of Phyllotaxis

 

Move your mouse over the purple dots to learn what each person added to our knowledge of phyllotaxis, or click here to see their accomplishments simply listed.

 

 

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

wrote Enquiry into Plants which mentions leaves in regular series.

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

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.

 

Ancient Egyptians

 

 

 

370-285 BC Theophrastus

 

 

 

23-79 AD Pliny

 

 

1202 Leonardo Fibonacci

 

 

1452-1519 Leonardo Da Vinci

anticipated, in one of notebooks, the observation of spiral patterns in plants such as those made by Bonnet.

 

 

 

1571-1630 Johannes Kepler

was fascinated by the number five 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 promordia 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.