Classic Geometric Models of Phyllotaxis

van Iterson

      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 helical lattices.

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