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 here.

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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 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.