Cover art

I wish I could take credit for designing the fractal cover art for this blog, but it was actually generated by a computer program written by a couple of my former Calculus I students who attended the Texas Academy of Mathematics and Science, a residential program at the University of North Texas for gifted high school juniors and seniors.

More specifically, the cover art depicts solutions found using Newton’s method (see Wikipedia or MathWorld for more details) for finding the roots of f(x) = x^6-1. There are six roots of this polynomial, and so Newton’s methodwill converge to one of these six roots when beginning at various points at the complex plane. However, the method does not necessarily converge to the closest root. The cover art depicts the roots that are found when beginning from various points in the complex plane, while the darkness of the color depicts the number of steps necessary before the method converges within a small distance \varepsilon of the root.

The cover art is a fractal because, in some places, a slight perturbation in the initial starting place can have a dramatic effect on the root ultimately found by Newton’s method. Indeed, if one were to zoom into the picture where the most colors are bunched together, the blown-up figure would look very much like the original figure.

Such sensitivity to initial conditions is colloquially known as chaos or the butterfly effect (again, see Wikipedia and MathWorld for more details).

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