A bisymmetry equation on restricted domain by Kocsis I.

By Kocsis I.

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People seem to think that because something involves mathematics it is necessary to know mathematics to use it. Radio does indeed involve sines and cosines, but the person adjusting the dials needs no trigonometry. Geologists searching for oil do not have to solve differential equations, though differential equations may have been involved in the creation of the tools that geologists use. I am not saying that mathematics is never required in the workplace. Of course it is, and it has helped to make our technology what it is.

Every couple minutes the screen would go blank and refresh itself with a completely different fractal. I have to confess that I spent a few idle minutes watching the fractals instead of writing. One day, a new design popped up on the screen (see the first figure). It was different from all the other fractals. It was made up of simple shapes— circles, in fact—and unlike all the other screen savers, it had numbers!  . They were the perfect squares! The sequence was 1 squared, 2 squared, 3 squared, and so on.

One of Apollonius’ lost manuscripts was called Tangencies. According to later commentators, Apollonius apparently solved the problem of drawing circles that are simultaneously tangent to three lines, or two lines and a circle, or two circles and a line, or three circles. The hardest case of all was the case where the three circles are tangent. No one knows, of course, what Apollonius’ solution was, or whether it was correct. After many of the writings of the ancient Greeks became available again to European scholars of the Renaissance, the unsolved “problem of Apollonius” became a great challenge.

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