By now everyone with an interest in science or computers has stared in wonder at pictorial representations of the amazing fractal called the Mandelbrot set. If you are one of the few remaining holdouts, then do yourself a favor and explore the full color illustrations of this infinite mathematical labyrinth (one readily available source is James Gleick’s popular book Chaos). These images, depicting seemingly endless layers of complexity and geometric inventiveness, strike deep intellectual and emotional chords. A large part of the magic is due to the fact that as you plunge deeper into the Mandelbrot set you encounter ever-more-tiny copies of itself within the riot of detail. This “worlds within worlds” aspect, or self-similarity, calls to mind the famous poem of William Blake:

“To see a World in a Grain of Sand…
And a Heaven in a Wild Flower
Hold Infinity in the Palm of your hand
And Eternity in an hour.”

A truly remarkable thing about the Mandelbrot set is that it is not generated by long strings of incomprehensible equations, but rather by a simple recursive algorithm that can be embodied in a few lines of computer code. So the seemingly infinite complexity of the Mandelbrot set has a simple underlying order. Here one is reminded of Coleridge and his “unity in variety.” Using this fractal as an archetype, one can say that two hallmarks of fractal systems are: (1) inherent hierarchical organization, and (2) self-similarity, i.e., the copies within copies within…motif…

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