May 2, 2014

Phyllotaxis with Mathematica

Phyllotaxis with Mathematica

Expanding on Fermat’s Spirals, this pretty depiction was generated using a simple mathematical expression and closely resembles a “sunflower”.

Employing the irrational number φ = 1.618…, known as the Golden Ratio, is the key to the forming this particular pattern. If the expansion of the opposing Fermat spirals grow proportionally to φ, from the center outwards, then an even packing of the circles is achieved, resulting in the shape you see here.

Nature has clearly discovered this space saving optimization. We see its equivalent all the time in daisies, sunflowers, pineapples, pinecones and more.

The Mathematica code used to generate these figure is provided below:

k = 1.;
d = 2*Pi/N[GoldenRatio];
f[t_] := k*Sqrt[t];
x[t_] := f[t]*Cos[t];
y[t_] := f[t]*Sin[t];
plotTo = 55.;
data = Table[
Graphics[{Hue[.17 – 0.00002*t],
Disk[{x[t], y[t]}, 0.8 + 0.0005*t]}], {t, 1, 2500, d}];
Show[data, PlotRange -> {{-plotTo, plotTo}, {-plotTo, plotTo}}]