Romanesco broccoli is a fantastic illustration of fractal geometry in nature. Fractals demonstrate the property of self-similarity. In a fractal, each part is a copy of the whole.
I remember how much more beautiful the sight of bare tree branches against a gray November sky became to me once I had learned to see them in terms of fractals. Trees in the winter provide a perfect opportunity for observing what Benoit Mandelbrot has called the fractal geometry of nature. Each large branch traces out the same basic pattern as the entire tree. Each smaller branch has the same shape as the larger branch and, in turn, the same shape as the entire tree. This sameness at all levels of organization from the smallest twig to the whole tree is what is known as self-similarity.
The fractal concept is the same as the principle behind a hologram, a type of laser-created image in which each part contains a copy of the whole. Ferns are another plant whose self-similar nature is easy to see. All along the long fern frond run rows of leaves, each leaf a perfect copy of the entire frond. And each of these leaves is composed, in turn, of smaller leaflets, each also a copy of the entire fern.
A lot of people, including me, find fractals to be very beautiful; their beauty is probably one of the main reasons I have studied them for so long. It is certainly what first attracted me to the subject. I once heard a talk about fractals in which the speaker postulated a role for fractals in determining what is beautiful to us as humans. He suggested that we develop our sense of beauty by being surrounded, on all sides, by fractals in the natural world and have evolved to appreciate their existence. Whether or not this explanation for the development of our sense of beauty is true, it is certainly true that fractals are very pleasing to the eye. Their inherent regularity (the pattern of repeating self-similarity) combined with an equally fundamental irregularity (the basic unit, usually quite random, which is repeated in the pattern) is very pleasing to us. Trees and ferns are fractals, but so are lightning strikes, frost crystals growing on a pane of glass -- even things in nature we cannot readily see such as bronchial tubes, blood vessel branchings and networks of brain cells. All of these fractal objects illustrate the principle of microcosm and macrocosm, in which the part is a perfect reflection of the whole.
The fractal concept comes up in nonlinear science in a deep and fundamental way: a strange attractor (i.e. one that stabilizes chaotic behavior) is a fractal object. A small portion of the attractor will, when enlarged, have the same basic pattern of trajectories as the whole. Only chaotic attractors have this property – steady state and limit cycle attractors don’t.
And there is a reverse connection as well. Take the Mandelbrot set, for example. This famous example of a fractal corresponds to a mathematical relationship that actually generates chaotic trajectories.
The main take-home lesson here is that chaos and fractals are two sides of the same coin. The type of dynamic relationships that lead to the stable but unpredictable behavior of chaos also lead to the self-similarity of fractals. I don't know about you, but it seems to me that the fact that unpredictable chaotic behavior can also lead to astounding beauty is profound commentary on the true nature of reality.