Spiral Lattices

Spiral lattices are configurations of points placed on concentric circles such that:
• There is one point on each circle.
• The divergence angle (d) between points on successive circles is constant.
• The radius of successive concentric circles increases by a constant factor G, the expansion parameter (also called Plastochrone Ratio).

The applet Spiral Lattices shows all possible spiral lattices, and points to their classification.

The coordinates of the kth point in a spiral lattice with divergence d and expansion parameter G are given by (G^kcos(kd), G^k sin(kd)). Note that these points all belong to a continuous spiral parametrized by (G^tcos(td), G^t sin(td)), where t is a real number. This spiral is called the generative spiral. The generative spiral is usually not the one that is visible: in most cases, it winds too much to catch the eye.
Certain patterns (eg. the head of a sunflower) are better approximated by (sqrt(kG)cos(kd), sqrt(kG) sin(kd)).

In a spiral lattice, the eye tends to connect nearest points into spirals. These spirals are called parastichies. There are two sets of parastichies winding in different directions. In the figure to the right there are 8 parastichies illustrated in red and 13 in grey. Spiral lattices are classified according to the number of parastichies in each set. The lattice shown here has parastichy numbers (8, 13).

Parastichy numbers are related to which points are closest to a given point. The nearest neighbor to point 0 is point 13. One of the grey parastichies in the figure passes through the points labeled 0, 13, 26 etc. The difference between the numbers on two neighboring points in this parastichy is always 13, which is the number of grey parastichies. This is true for all of the grey parastichies.

Similar statements hold for the red parastichies. The next nearest neighbor to point 0 is point 8. Points 0 and 8 lie on a red parastichy and 8 - 0 = 8. The next nearest neighbor to point 7 is point 15. Points 7 and 15 lie on a red parastichy and 15 - 7 = 8.

Each spiral lattices can be seen as a discrete subgroup of the complex multiplicative group, with the generator Ge^id. Indeed, in complex notation, points of the lattice can be written as G^ke^ikd. These form a group isomorphic to the integers (the isomorphism is k-> G^ke^ikd). Parastichies correspond to cosets of the subgroups 8Z and 13Z in the example given here.