Recursion in Nature and Art

Our first piece of recursive code will be a drawing a logarithmic spiral. It was first described by Descartes. Jacob Bernoulli was so impressed by it that he asked for it to be engraved on his tombstone. He named it spira mirabilis, the miraculous spiral. Here are examples of this spiral occurring in nature:

This spiral is also the path taken by a charged particle moving perpendicular to a uniform magnetic field. It is also the path taken by hawk honing in towards its prey. Here is the code. In this code we have a step size, a decay factor that shortens the step size, and an angle the turtle turns after every drawing of a step. The three parameters - step size, decay factor, and angle can be changed to match observation.

We will examine some traditional recursive structures. We will be drawing some well known fractals. A fractal has the same basic form now matter how much you magnify it. In nature you can find fractals in the form of outlines of clouds and coastlines, snowflakes, mountains, and even trees.

Sierpinski Gasket or Triangle
We start with an equilateral triangle and join the mid points of each side. This forms four smaller triangles. For each of the three outer triangles we join the mid points of their sides and so on.
Fractal Tree
We define a tree as having a trunk and a smaller tree going off to the right another smaller tree going off to the left. And then we apply the same definition to both of the smaller right and left trees.
H Tree
To draw a H-Tree of order n, we use three lines in the shape of the letter H. Then we add four H-trees of order n-1, connected to each of the four tips of the original H. Each time the size of the H-tree is halved.
Hilbert Curves
A Hilbert curve is a space filling curve that is a continuous curve that passes through every unit square in a grid. We start with a basic shape and the rest of the curve is generated by reproducing that basic shape at various scales. Here is a short tutorial on Hilbert curves. You may also want to look at the link on other space filling curves like the Peano curves.
Koch Snowflake
The Koch snowflake is created as follows: Begin with an equilateral triangle. Divide each side of the equilateral triangle into three equal segments. On the middle segment draw an outward equilateral triangle. Repeat the process for higher order snowflake.
Mandelbrot Sets
IBM mathematician, Benoit Mandelbrot, popularized the term fractal in the early eighties. What is fascinating about the set is that the boundary contains numerous circular shapes, each of which contains other circular shapes. The set is generated by an iterated function system that involves compex numbers.

M. C. Escher (1898 - 1972)

Piet Mondrian (1872 - 1944)

In most of the drawings that we did, we used straight line segments. Here is an iterative code that uses very small straight line segments to draw arcs. It then uses these arcs to form a Sun like figure.

Recursion in Music

It turns out that musical rhythm is recursive. There are self-similar patterns that repeat in music. Each composer has his own rhythmic signature. Listen to this NPR News Item.