Classic Geometric Models of Phyllotaxis
van Iterson performed a geometric analysis almost thirty years
after Schwendener. van Iterson combined the approaches of Airy and
Schwendener by considering the packing of spheres in a various
arrangements in space such as helical lattice patterns. When the
diameter of the spheres is smaller than the diameter of the helix
the sphere packing can be seen as taking place on the surface of a
cylinder. This is shown in van Iterson's figure 11 on the left.
When the sphere's diamter is larger than the diameter of the helix
there can be no underlying cylinder. This is shown in van
Itseron's figure 4 on the right.
van Iterson's approach was a bit more mathematical than Schwendener's
although his goal was the same - the study of leaf arrangements.
van Iterson recognized that the whole set of helical lattices could be
parameterized by two numbers - the divergence angle and internode.
The divergence angle is the angle between successive lattices points
and the internode is the height between successive lattice points
van Iterson recognized that when the spheres were packed along
rhombic lattice patterns the contacts between the spheres followed
the parastichies of the lattice. van Iterson computed the subset of
rhombic lattices within the space of helical lattices and produced
what is now known as the "van Iterson diagram". This is shown below.
The original van Iterson diagram.
The horizontal axis is the divergence angle labeled in degrees. The
vertical axis is the internode represented as a proportion of the
cylinder's circumference. The set of divergence angle, internode
pairs which form rhombic lattices lie on semi-circles whose end points
sit on the horizontal axis. van Iterson drew the set of those rhombic
for which the interior angles of the rhombic unit cell are between
60 and 120 degrees as solid curves. We call these the "fat" rhombic
lattices. The remaining rhombic lattices are called "thin". van
Iterson draw these with dotted curves.
The fat rhombic lattices with parastcihy numbers (1,1) actually
sit on a vertical line. They have divergence angle equal to 180
degrees and internode from the square root of 3/4 (~0.866) to the
square root of 1/12 (~0.289).
The helical lattices with divergence angles between 180 and 360
degrees are the mirror image of the helical lattices with divergence
angles betwen 0 and 180 degrees. Below we show van Iterson's diagram
with a copy obtained by reflecting the original about the right hand vertical side. This shows the full set of rhombic lattices.
van Iterson's diagram with its mirror image
Above we see the fixed point bifurcation diagram of the dynamical
model. This is shown with the black curves. It is a subset of the
van Iterson diagram.
The set of semi-circular arcs perpendicular to the horizontal axis and
joined with the set of vertical half lines perpendicular to the
horizontal axis constitute the lines in one of Poincare's models of
hyperbolic geometry. Thus the set of rhombic lattices can be
thought of as a collection of lines in hyperbolic geometry.
The set of thin rhombic lattices form the boundaries between regions
with different parastichy numbers. Partitioning the plane into
regions of constant parastichy numbers produces a tiling of the
hyperbolic plane into two types of tilings. One type of tile is
hexagonal and the other is an infinite sided polygon. The tiling is
symmetrical under a subgroup of the unimodular group and so its
structure is esstially the same everywhere. This is shown below.
Tiling of the hyperbolic plane according to parastcihy numbers of