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 distance)
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. |