Higher frequency (φ + n) phyllotaxis patterns

A phyllotaxis pattern for φ + 20.

Phyllotaxis patterns are often described that have a frequency (i.e., number of dots per turn of the spiral) that is either phi (φ) or phi squared—which is identically phi + 1—but a frequency that is φ plus a larger integer will also work in certain cases. In the plot above, the frequency is phi + 20 dots per turn. A central zone of the pattern does not form a circle packing, but this zone would not be needed in a heliostat array since the central optics would be located there. 

A processing sketch was used to generate these patterns.

Circles packed along a φ + 10 phyllotaxis spiral.

Circles packed along a φ + 20 phyllotaxis spiral.

Circles packed along a φ + 50 phyllotaxis spiral.

These patterns also work with radially oriented ellipses:

36:1 ellipses packed along a φ + 10 phyllotaxis spiral.

36:1 ellipses packed along a φ + 20 phyllotaxis spiral.

36:1 ellipses packed along a φ + 50 phyllotaxis spiral.