The complex plane and the cyclamen leaf

A couple of years ago, for New Year's Eve, I went with my friends in the Foreste Casentinesi National Park, one of the wildest places in Tuscany: ancient forests, deep valleys and beautiful creeks, places that Dante used in his Commedia to describe a wild river in the Inferno.
I was there with some friends and we were exploring the oak woods near a beautiful waterfall. There I found the leaf.


It's the leaf of a cyclamen, a pretty flower that grows in our forests during winter. It's a very common view, though it is a protected species. But still... I suddenly stopped, staring at it. Then I took its photo.
What mesmerized me was the shape on the leaf, something that anyone interested in math, computer graphics or at least colourful desktop wallpapers will surely recognize. It is a fascinating (yet a little creepy) natural reproduction of the graphical representation of a Mandelbrot Set, perhaps the most famous fractal outside the world of mathematicians.


What is a fractal? According to Benoît Mandelbrot, who described them in 1975, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole". In one word, recursivity. These mathematical figures are usually built from relatively simple recursive, iterated algorithms. Even Mandelbrot equation is (or seems...) incredibly simple:

Z=Z²+C,

where C is a complex parameter. You take a point C on the complex plane and then test it, iterating the equation while changing the value of Z starting from 0. If Z (or better its magnitude) remains equal to or below 2, that point belongs to the set. If not, the point is not part of the set. The black area in the figure represents all the points belonging to the set. Colours of the points outside the sets are obtained calculating how many iteractions are required to surpass 2.
I won't go any further into it, so I suggest the Wikipedia page or, better, this introductive guide to the Mandelbrot set.

Nature prefers fractal over Euclidean geometry. Many natural shapes are recursive: a river with all its tributaries, a mountain range, the feather of a bird and then an endless series of fascinating examples among plants, like the branches of a tree, the leaves of a fern, the flower of a romanesco broccoli (broccolo romano, please) and so on. Recursive algorithms may create very "natural looking" shapes (like the computed fern in figure), especially if a certain amount of randomness is introduced in the calculation.


Among many cultures, especially those more "bound to nature", the mental representation of visual space is built upon fractals, like among the Maasai in Kenya and Tanzania: their villages are circles of circular huts (the boma) in which objects are arranged in circles. A simple task like aligning things parallel may not be immediately understood.

Thus, it's not a surprise that a cyclamen leaf has a fractal decoration. Still, it was a dizzy, impressive view: after all, the figure is the representation of an equation involving imaginary numbers.