Cylinder Lattices Applet
This applet shows all the possible
(half) cylindrical lattices. Think of the right window as
a vertical cylinder (e.g. the stem of a plant) which has been
unrolled. In terms of botany, the vertical axis is the internodal
distance, the horizontal axis is the divergence angle. Note
that, by symmetry, only positive divergence angles are represented
How To Use This Applet
Choose a generator for the lattice in
the configuration space by moving the (clicked in) mouse in the
parameter space. The other generator is fixed, and horizontal, of
length equal to the circumference of the cylinder. The number on
the top are the x and y coordinates (divergence angle and internodal
Click on partition to
see the regions where the 2 closest neighbor to a point change.
Click on nearest point
to see the 2 closest points to the zero
th generation (central point on the bottom).
Click on numbers
to see the generations of the 2 closest
points to zero (the option nearest point has to be on). These numbers
also correspond to the parastichy numbers: the number of visible
helixes winding up the cylinder. Click Parastichies
to see a pair of these helixes (one in each direction).
Click on partition to
see the places where the second closest neighbor changes (black
curves). The regions these curves bound are regions of constant
parastichy numbers. These regions give an illustration of the fundamental
theorem of Phyllotaxis: the relationship between divergence angle
and parastichy numbers. The gray curves are where the first and
second closest points swap roles.
Click on circles to
visualize the successive generations of primordia (the circles appear
as horizontal lines on this unrolled cylinder).
Click on voronoi to
visualize the Voronoi cell decomposition of the lattice (the Voronoi
cell of a lattice point P is the region of all the points in the
plane that are closest to P than to any other lattice point).
Click on potential to
see the bifurcation diagram of an "ideal" repelling potential: the
derivative of the repelling potential is negative at 0 for lattices
in the white region, and positive in the ochre region. Hence local
mins are attained at points which bound these regions, with the
white on their left. As you come down from a high internodal distance,
the only possible connected path of minima of the potential meanders
through regions of
Click on zoom to
focus on the parameter region bounded by the green square.