  Expanding on the Fermat’s spiral example, these pretty depictions look like “daisy” patterns. Mathematically the two plots are the same, the only difference being the left plot uses circles and the right one diamonds.

The use of the irrational number φ (Phi = 1.618…), known as the Golden Ratio, is integral to the formation of this pattern, which can readily be observed in nature, for example, in daisies, sunflowers, pineapples and pinecones.

As it turns out, if the expansion of the opposing spirals from the center outwards grow proportionally to φ, then an even packing of the circles or diamonds is achieved, resulting in the pattern you see here. Nature has clearly discovered this space saving optimization.

The Mathematica code used to generate these figures is provided below.

(*Using Circles*)

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}}]

(*Using Diamonds*)

(*Spiral*)

k = 1.;
d = 2*Pi/N[GoldenRatio];
f[t_] := k*Sqrt[t];
x[t_] := f[t]*Cos[t];
y[t_] := f[t]*Sin[t];

(*Diamond*)

c = 5.;
d = 3;
diamond[xx_, yy_, rot_, s_, h_] := Graphics[Rotate[{Hue[h], Polygon[
{{xx, yy + s*c},
{xx + s*d, yy},
{xx, yy – s*c},
{xx – s*d, yy}}]}, -1*(Pi/2 – rot)]];

(*Table of Graphics objects mapped as spirals*)

data = Table[Graphics[diamond[x[t], y[t],
ArcTan[y[t]/x[t]],
(1 + 0.00025*t),
(0.15 – 0.00002*t)]],
{t, 1, 2500, d}];

(*Plot all Graphic objects together*)

plotTo = 60.;
Show[data, PlotRange -> {{-plotTo, plotTo}, {-plotTo, plotTo}}]