## Fractals

If you examine a bifurcation diagram closely, you begin to see interesting patterns. For example, start with a completed diagram, such as the one in the first picture.

Next, zoom in on the first doubling point. It looks like a rounded, sideways V. Now look at the smaller, sideways V's that come next in the series.

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Now zoom in again, say, on that upper, smaller V.

Notice how this region of the diagram looks like the original. In other words, the large-scale structure of the figure is repeated multiple times. The doubling regions exhibit a quality known as **self-similarity** -- small regions resemble large ones. Even if you look in the chaotic areas of the diagram (which occur to the right), you can find this quality.

Self-similarity is a property of a class of geometric objects known as **fractals**. The Polish-born mathematician Benoît Mandelbrot coined the term in 1975, after the Latin word *fractus*, which means "broken" or "fragmented." He also worked out the basic math of the objects and described their properties. In addition to self-similarity, fractals also possess something known as **fractal dimension**, a measure of their complexity. The dimension is not an integer -- 1, 2, 3 -- but a fraction. For example, a fractal line has a dimension between 1 and 2.

The **Koch snowflake** -- named after the Swedish mathematician Helge van Koch -- stands as a classic example of a fractal. To derive the shape, van Koch established the following rules, first for a line:

- Divide a line segment into three equal parts
- Remove one-third of the segment out of the middle
- Replace the middle segment with two segments of the same length such that they all connect
- Repeat indefinitely on each line segment

The second picture shows what the first two iterations would look like:

If you start with an equilateral triangle and repeat the procedure, you end up with a snowflake that has a finite area and an infinite perimeter:

Today, fractals form part of the visual identity of chaos. As infinitely complex objects that are self-similar across all scales, they represent dynamical systems in all their glory. In fact Mandelbrot eventually proved that Lorenz's attractor was a fractal, as are most strange attractors. And they're not limited to the ruminations of scientists or the renderings of computers.

Fractals are found throughout nature -- in coastlines, seashells, rivers, clouds, snowflakes and tree bark. Before you take a field trip, however, be aware that self-similarity behaves a little differently in natural systems. In controlled mathematical environments, an object with self-similarity often displays an exact repetition of patterns at different magnifications. In nature, patterns obey statistical self-similarity -- they don't repeat exactly but parts of them show the same statistical properties at many different scales.