

Hofmeister's
Rules  Dynamical System Model  Results



A Mathematical Model
We present the dynamical system model of phyllotaxis which has
been the object of our research.
It is a mathematical derivation of a model by the physicists Douady
and Couder, which itself is based on observations by the 19th Century
botanist Hofmeister. This model reproduces all spiral patterns exhibited
by plants and explains why Fibonacci phyllotaxis
is predominant. The applet Dynamical
Model gives a simulation of this process.



Hofmeister's Rules 
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In 1868 the botanist Wilhelm Hofmeister published
his microscopic study of meristems.
He proposed that in spiral phyllotaxis, botanical units form one
by one with the newest in the least crowded spot. Following Douady
and Couder (1996, Part I), we make these rules a bit more precise:
• Primordia initiate one
at a time in the least crowded spot around a circular meristem.
• Primordia are radially displaced away from the center
as they grow.









Compare a scanning electron micrograph of a Norway
spruce (Picea abies) shoot apical
meristem with a computer generated arrangement of primordia. The
computer program implemented the mathematical model based on Hofmeister's
rules and converged naturally to this configuration. Primordia are
numbered according to their age  the higher the number, the older
the primordium. Notice the remarkable similarity between plant and
model.
Electron micrograph courtesy of Rolf Rutishauser, published in Rutishauser
(1998).




Dynamical System Model 
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Our model is a discrete dynamical system. Generally,
a discrete dynamical system consists of iterating a function (also
called map) F on a set S (also called the state
space). In this model, each element in S corresponds to
a configuration of primordia, which are represented by points on
concentric circles, with one point per circle. The innermost circle
represents the periphery of the meristem. From circle to circle
the radius increases by a constant factor G, that we call
here expansion rate (Plastochrone ratio in the
literature). Each configuration is encoded as a list of divergence
angles between successive points. Hence, S can be seen as the set
of all possible lists (d_{1} , d_{2} , . . .
,d_{N}) of angles.









Under the function F, points of a configuration
move out radially to the next circle. The outermost point is eliminated
and a new point (green) is generated on the innermost circle, where
the distance to the closest point is maximum. The "distancetoclosest"
function along the inner circle is represented as a graph (red).
the "distancetoclosest" function D(z), with
variable z representing a point on the edge of the meristem
is implemented as D(z) = min_{k}(dis(z, z_{k}))
where z_{k} is the point representing the
center of primordium k. Clearly each z_{k }is
function of the variable d_{1} , d_{2} , . .
. ,d_{N}. The least crowded rule chooses the point
z that maximizes D.




Results 
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Stationary points in Dynamical Systems The
first feature that one usually studies in a dynamical system is
its fixed, or stationary points. These
are points in the state space which do not change under the map
F. For example, the dynamical system given by squaring
real numbers, f(x)=x^{2} has fixed points 0 and
1, since f(0)=0 and f(1)=1. An important characteristic
of fixed points is their stability. In the example
of the squaring map, 0 is stable: if you start with a point close
to 0 (e.g. 0.01), repeated squaring will only get you closer
to 0. On the other hand, the fixed point 1 is unstable: repeated
squaring of numbers greater than 1 will indefinitely increase these
numbers, whereas repeated squaring of numbers smaller than 1 will
decrease to 0; in either case, you will get farther and farther
from 1. As a rule of thumb, if the dynamical system considered models
a natural phenomenon, stable fixed points are significant in the
sense that they can be perceived in experiments, as their stability
allows for a margin of error in measurements.
Stationary configurations in the Phyllotaxis Model
In the model F described above, we have the following results
(see Hotton (1999) and Atela,
Golé & Hotton (2002)):
 Stationary configurations of F are spiral
lattices, i.e. configurations of constant divergence
angles d_{1} = d_{2}
= . . . =d_{N } (a trivial consequence of
the rule of radial displacement).
 All stationary configurations of F are stable, accounting
for the presence of spiral phyllotaxis in nature.
 Among all spiral lattices (each represented by a point of the
disk below, see the Spiral
Applet), those corresponding to stationary configurations
form a collection of paths traversing regions of lattices with
parastichy numbers following Fibonaccilike sequences. The two
most prominent paths start, for large values of the expansion
parameter G (represented by points closer to the center
below), at the (1,1) (distichous) lattices (the red line
segment starting leftward from the center), and follow regions
of increasing Fibonacci numbers as G decreases.








A Scenario for Fibonacci Phyllotaxis These
three results put together provide given Hofmeister's rules an
explanation for the occurrence of spiral phyllotaxis in nature,
as well as for the predominance of Fibonacci phyllotaxis in spiral
phyllotaxis. Indeed, the expansion parameter G is known
to decrease as plants grow, especially at the flowering stage. With
large G the pattern has no choice but to converge toward
a distichous phyllotaxis. As G decreases, and primordia
are formed (i.e.. while the dynamical system is iterated), the stability
of stationary configurations forces the nearby configurations to
hover along one of the paths of fixed points which starts at distichous,
that is, one of the two Fibonacci paths. As G decreases
(moving away from the center in the picture), parastichy numbers
increase along the Fibonacci sequence, and the divergence angle
tend the Golden angle (upper path) or its complement (lower path).
Such a scenario is numerically  and visually  verified in the
Dynamical Model
applet.
Hofmeister's Rules  Dynamical
System Model  Results 


