The word “fractal” was coined by Benoit Mandelbrot (namesake of both the Mandelbrot set and a certain hard-nosed private eye) in 1975. Mandelbrot was not the first mathematician to examine these perplexing shapes, but he launched the formal study of them as a scientific tool to understand complexities found in nature. His book, The Fractal Geometry of Nature paved the way in this regard.
For a straightforward, illustrated guide to fractals, you can check out Introducing Fractal Geometry.
NOVA documentary about fractals that includes interviews with Benoit Mandelbrot.
A great explanation of how the Mandelbrot set is generated.
You can use this program to generate and explore fractals.
A music video rendition of Jonathan Coulton’s Mandelbrot Set
A more complete definition of fractals than the one found in this video is:
- Fractals are generated through iteration (repeating the same operation infinitely)
- Fractals display self similarity (they’re repeat the same patterns no matter how much you zoom in)
- Fractals have fractional Hausdorff dimension. Essentially, this means that they’re rougher than platonic solids—more jagged and with fewer smooth, predictable curves. If you were to walk through a thicket of fractals, they’d stick to your clothes a lot more than spheres and cubes would.
This Wikipedia page lists quite a few fractals in increasing order of Hausdorff dimension.
Imagine a two-dimensional world -- you, your friends, everything is 2D. In his 1884 novella, Edwin Abbott invented this world and called it Flatland. Alex Rosenthal and George Zaidan take the premise of Flatland one dimension further, imploring us to consider how we would see dimensions different from our own and why the exploration just may be worth it.
Benoit Mandelbrot, namesake of the Mandelbrot set and all-around fractal pioneer, observed that “clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” In fact, fractals can be found all over the natural world—some examples include the nautilus shell, Romanesco Broccoli, and the coastline of Great Britain.