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 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.
