This program, written at different stages by Scott Hotton (UCSC), Pau Atela, Chris Golé and Biliana Kaneva (Smith College), explores the universe of spiral lattices. The program shows all mathematically possible spiral lattices and the underlying structure that classifies them. It then helps the user identify those lattices that are favored by plants (Fibonacci phyllotaxis). You can can stay on this page and follow the 7 steps of our tutorial or you can go back to the applet for a quick start).

It will take us seven steps to explore the program and get a glimpse of what Phyllotaxis is about.

When you first view the Spiral program, you see 2 panels inside a central frame.
The left one is the *Parameter window*, the right one is the *Configuration
window *. When you point and click the mouse inside the circle of the parameter
window, the program draws a set of points outside the circle in the configuration
window. Keeping the mouse button pressed and moving, you can explore all the
possible spiral lattices (go to the Spiral Program).

The lattice with expansion rate *G * and
divergence angle *d *is represented by the point of polar coordinates
(1/*G*, -*d*) in the parameter space. In particular, lattices
near the center have large *G, *those near the boundary of the disk have
small *G. *In complex notation, if the mouse points at the point z of
the unit disk in the parameter space, the points in the spiral lattice drawn
in the Configurations window are the positive integer powers of 1/z.

When you bring the mouse closer to the edge of the circle, you may notice other spirals than the generative spiral: Your eyes are trying to interpolate curves between points that are close to one another. A careful analysis of which points are closest to a given one is the key to the structure of this program (and of phyllotaxis). A point in a configuration has 1, 2 or (more rarely) 3 outbound closest neighbours. Press on the button "closest points" in the control box. The two closest points to the point (1,0) (in blue, at the edge of the circle) appear in red (closest) and green (2nd closest).

Now press successively on "Spirals" and "All spirals". The red spirals join
each point to its closest neighbours. The green spirals join the second closest
neighbours. Presumably, these are the spirals that your eyes perceived.In phyllotaxis,
these spirals are called *Parastichies*.

As you move the mouse with the "All spiral" option on, you may notice qualitative changes. These changes are of two types:

- Change of type 1: the spirals interchange colours
- Change of type 2: the green spirals change altogether. You may want to count the red and green spiral before and after a change of type 2.
- Change of type 3: the green spirals disappear, only the red ones remain.

A change of type 1 occurs when the closest and second closest points to (1,0)
(or any other point in the configuration) are at same distance from (1,0). To
explore this, turn off the "Spirals" and "All spirals" options and turn on the
"Circle" option. A grey circle centered at (1,0) and passing through the green
point (second closest to (1,0)) appears in the parameter space. Moving the mouse
in the parameter space, find a locus of change of type 1 finding spots where
the red point crosses the circle (and then becomes the green one).

A change of type 2 occurs when a third point "dethrones" the currant second
closest. Find the locus of such a change with the "circle" option on: look for
points outside the circle coming into it to become the new second closest.

A change of type 3 occurs when the second closest point is in the same spiral
as the one generated by the closest one: the green spirals are also red. By
default, the programs paints them red.

Note that only at changes of type 2 or 3 can there be a change in the number of visible spirals. The locus of all the possible changes of type 2 is made of all the white curves drawn in the parameter space when you turn on the "Picture" option. Verify that the number of green spirals indeed changes when the mouse crosses such a line.

The blue regions represent values of the parameter for which green and red spirals are distinct. In the yellow region of the parameter space, the green spirals are also red. Hence the boundaries between blue and yellow regions form the locus of the changes of type 3 (go to the Spiral Program).

To see what configurations plants overwhelmingly choose among all possible
configurations, try to locate the regions of the parameter space where there
are 1 red, 1 green spiral, then 1 red, 2 green, then 1 green, 2 red , then 3G,
2R, then 3R, 2G, then 3R, 5G etc... following the Fibonacci sequence. You will
then note the following:

1) the parameter point approaches the boundary of the circle at a point whose
angle with the x-axis is either 137.5 degrees or 360-137.5=222.5 degrees.

2) you will have followed a trajectory which is not too far from a red line
drawn in the parameter space.

The points in the red curves of the parameter space picture correspond to
stationary configurations of the dynamical system.
This set of red curves is called the Bifurcation Diagram. The most prominent
curves in the Bifurcation Diagram as we saw above, are those whose parastichy
numbers follow the Fibonacci sequence. They emerge from the set of distichous
lattices near the center of the disk, where the expansion rate *G* is
large. As one moves along one of these curves away from the center (as *G
* decreases), one is lead through regions of lattices with Fibonacci phyllotaxis.
The fact that all these stationary solutions are stable gives a plausible scenario
for the emergence of Fibonacci phyllotaxis: the expansion parameter *G *(called
plastochrone ratio by botanists) is known to decrease in plants, especially
at inflorescence.