6.6 Growth and Movements of Plants
101
Arrangements of growing leaves and other iterative phylla have long fascinated
both natural scientists and mathematicians. The spacing of primordia is determined
by the inhibitory action of existing leaves, which is reminiscent of Turing’s symmetry breaking mechanism, although it was realized by botanists (Schoute, 1913) long
before Turing. An intriguing feature is the pattern of two mutually intersecting spirals, called parastichy (Fig. 6.16, center and right), related by Fibonacci numbers.
This classical sequence, originating from the counting of breeding rabbits in a book
written by Leonardo of Pisa in 1202 (known by his patronimic, filius Bonacci), but
known in India perhaps as early as the 5th century BC, starts with 1 and 1, while
each subsequent member of the sequence is the sum of the two preceding ones: 1,
1, 2, 3, 5, 8, 13, etc. What has this got to do with plants? Douady and Couder (1992)
constructed spiral patterns governed by the Fibonacci sequence with the help of a
model so simple that it can be explained to a lay reader, and implemented it both by
simulations and by experiments bearing no relation to plants.
Botanists had come upon the empirical rule that a new primordium appears with
periodicity T near the tip in the largest gap left between the previous primordia and
the apex. Couder and Douady assumed that, due to inhibition, new phylla nucleate
at a distance R from the apex as far as possible from the preceding ones and are
advected from the center due to growth at a velocity V . In this model, all relevant
distances should be proportional to R and all times to T , so that these values can be
used as length and time units. Accordingly, R/T can be taken as the velocity unit.
The resulting pattern should depend only on the dimensionless rate of growth V .
Let the first primordium appear at time 0 to the east of the tip, i.e., at an angle
φ = 0. The second should then appear at time 1 on the west side, i.e., φ = π (180 ◦ ).
At time 2, the first outgrowth will be at distance 3 from the center, and the second at
distance 2, both retaining their angular positions. When growth is very fast, V ≥ 2,
each new primordium is repelled only by the previous one, so that successive dots
move away in opposite directions. In the interval 2 > V ≥ 1, the angular position of
a new primordium depends on the two preceding ones, at 1 > V ≥ 2/3, on three, at
2/3 > V ≥ 1/2, on four, etc., at decreasing intervals. As the growth rate decreases,
the phylla form a spiral pattern with the number of branches increasing step by step
as the growth rate decreases.
Couder and Douady used a dynamic computation minimizing an “energy” dependent on the repulsion strength to compute spiral patterns, but it can be done in
a simpler way by determining the locations farthest from already existing phylla at
each step. Miraculously, the pattern advances along the Fibonacci sequence with
decreasing V in such a way that sharp transitions occur precisely at the abovementioned values of the growth rate where the number of phylla influencing nucleations changes. For V ≥ 2, phylla lie on two straight lines going in opposite
directions from the center; any two consecutive phylla, labelled by their nucleation
time, can be connected by an arc between these two lines, and the labels of consecutive phylla lying on the same line differ by two. Recall first the Fibonacci numbers,
1 and 2. Below V = 2, a true pattern of intersecting spirals forms; along one of
them, the labels of the consecutive phylla differ by 3, and on the other, by 5 – the
next members of the Fibonacci sequence. In a denser spiral pattern with V < 1, the
101
Arrangements of growing leaves and other iterative phylla have long fascinated
both natural scientists and mathematicians. The spacing of primordia is determined
by the inhibitory action of existing leaves, which is reminiscent of Turing’s symmetry breaking mechanism, although it was realized by botanists (Schoute, 1913) long
before Turing. An intriguing feature is the pattern of two mutually intersecting spirals, called parastichy (Fig. 6.16, center and right), related by Fibonacci numbers.
This classical sequence, originating from the counting of breeding rabbits in a book
written by Leonardo of Pisa in 1202 (known by his patronimic, filius Bonacci), but
known in India perhaps as early as the 5th century BC, starts with 1 and 1, while
each subsequent member of the sequence is the sum of the two preceding ones: 1,
1, 2, 3, 5, 8, 13, etc. What has this got to do with plants? Douady and Couder (1992)
constructed spiral patterns governed by the Fibonacci sequence with the help of a
model so simple that it can be explained to a lay reader, and implemented it both by
simulations and by experiments bearing no relation to plants.
Botanists had come upon the empirical rule that a new primordium appears with
periodicity T near the tip in the largest gap left between the previous primordia and
the apex. Couder and Douady assumed that, due to inhibition, new phylla nucleate
at a distance R from the apex as far as possible from the preceding ones and are
advected from the center due to growth at a velocity V . In this model, all relevant
distances should be proportional to R and all times to T , so that these values can be
used as length and time units. Accordingly, R/T can be taken as the velocity unit.
The resulting pattern should depend only on the dimensionless rate of growth V .
Let the first primordium appear at time 0 to the east of the tip, i.e., at an angle
φ = 0. The second should then appear at time 1 on the west side, i.e., φ = π (180 ◦ ).
At time 2, the first outgrowth will be at distance 3 from the center, and the second at
distance 2, both retaining their angular positions. When growth is very fast, V ≥ 2,
each new primordium is repelled only by the previous one, so that successive dots
move away in opposite directions. In the interval 2 > V ≥ 1, the angular position of
a new primordium depends on the two preceding ones, at 1 > V ≥ 2/3, on three, at
2/3 > V ≥ 1/2, on four, etc., at decreasing intervals. As the growth rate decreases,
the phylla form a spiral pattern with the number of branches increasing step by step
as the growth rate decreases.
Couder and Douady used a dynamic computation minimizing an “energy” dependent on the repulsion strength to compute spiral patterns, but it can be done in
a simpler way by determining the locations farthest from already existing phylla at
each step. Miraculously, the pattern advances along the Fibonacci sequence with
decreasing V in such a way that sharp transitions occur precisely at the abovementioned values of the growth rate where the number of phylla influencing nucleations changes. For V ≥ 2, phylla lie on two straight lines going in opposite
directions from the center; any two consecutive phylla, labelled by their nucleation
time, can be connected by an arc between these two lines, and the labels of consecutive phylla lying on the same line differ by two. Recall first the Fibonacci numbers,
1 and 2. Below V = 2, a true pattern of intersecting spirals forms; along one of
them, the labels of the consecutive phylla differ by 3, and on the other, by 5 – the
next members of the Fibonacci sequence. In a denser spiral pattern with V < 1, the