A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are "the same from near as from far". Fractals may be exactly the same at every scale or they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.
The Mandelbrot set is a mathematical set of points whose boundary is a distinctive and easily recognizable two-dimensional fractal shape. The set is closely related to Circular Julia sets (which include similarly complex shapes), and is named after the mathematician Benoit Mandelbrot, who studied and popularized it.
Images of the Mandelbrot set are made by taking numbers on the complex plane, calculating whether it tends to infinity when the formula is iterated on the number, then using the number as X and Y coordinates in the picture and coloring the pixel depending on whether it tends to infinity or not.
Highlighting ever-changing, spiraling colors and shapes on the sphere get students excited to learn more about calculus. They are also great for parties under the sphere!