Why Fractals Are So Beautiful

The world is full of beautiful geometry.

It’s something we start teaching our youngest children. This daisy is a circle. This dandelion is a sphere. And we repeat it all the way through high school: Honeybees build their hives in hexagons. Solutions to quadratic equations can be graphed as parabolas. Rates of change are found in the slope of lines tangent to a curve.

“God has established nothing without geometrical beauty,” astronomer Johannes Kepler wrote. Before him, planetary orbits were thought to be perfect circles. He discovered that they weren’t—but that they were in fact ellipses, still part of the same wonderful family known as Euclidean geometry, named for the Greek mathematician who wrote Elements around the year 300 B.C.

Kepler, Newton, Leibniz, Descartes, and others looked at the world and found time and time again that the Euclidean model accurately describes all kinds of shapes and events in nature.

Except when it doesn’t.

You don’t have to look hard to notice aspects of nature that clearly don’t fit the Euclidean framework. Rivers, mountains, coastlines, lightning, our circulatory system: Where’s the symmetry and structure? Where’s the order?

The answer, as mathematicians are discovering more and more often, involves fractals: geometric figures that occur in nature, even in seemingly chaotic systems.

But fractals aren’t easy to understand. Benoit Mandelbrot, who coined the term fractal in 1975, summed them up as “beautiful, damn hard, and increasingly useful.”

Repeat. Repeat. Repeat.

Relative to Euclid’s geometry, 1975 might as well have been last Tuesday. Mathematicians agree that we’ve only just ...

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