I never liked the image associated with Mandelbrot Sets. It always seemed creepy and flat. But, I just now stumbled across a gif that made it's practical application much more clear...
Example of the classic representation:
"A mathematician's depiction of the Mandelbrot set M. A point c is colored black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of c, respectively."
Here is the gif that blew my mind:
"With {z_{n}} iterates plotted on the vertical axis, the Mandelbrot set can be seen to bifurcate where the set is finite" Link
What it made me think of:
Suddenly Mandelbrot's infinite abstractions seem more attainable visualized as the plump arms of cacti.
For the curious:
Mandelbrot Set: "Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. In other words, the boundary of the Mandelbrot set is a fractal curve. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.
The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization and mathematical beauty."
- Matt