Thus, for example, the golden ratio spiral of branches on a plant stem can be viewed as a marvellous way to minimize the shading of leaves, while the elaborate patterns on certain mollusc shells can be viewed as marvellous ways to confuse the visual systems of supposed predators.

For h = GoldenRatio the substitution system is {0  {1}, 1  {1, 0}} (see page 890 ), for h = √ 2 it is {0  {0, 1}, 1  {0, 1, 0}} (see page 892 ) and for h = √ 3 it is {0  {1, 1, 0}, 1  {1, 1, 0, 1}} .

For case (c), λ is GoldenRatio , or (1 + √ 5 )/2 .

These included for example symmetry, the golden ratio, spirals, vortices, minimal surfaces, branching patterns, and—since the 1980s—fractals.

(In some cases it works to look at whether the limiting ratio of lengths on successive steps is a Pisot number.) … The spectrum is roughly like the markings on a ruler that is recursively divided into {GoldenRatio, 1} pieces.

The number of sequences s n of length n that can actually occur is given by Apply[Plus, Flatten[MatrixPower[m, n]]] where the adjacency matrix m is given by MapAt[(1 + #) &, Table[0, {Length[net]}, {Length[net]}], Flatten[MapIndexed[{First[#2], Last[#1]} &, net, {2}], 1]] For rule 32, for example, s n turns out to be Fibonacci[n + 3] , so that for large n it is approximately GoldenRatio n .