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M.C. Escher - Art, Mathematics, Recursion, Infinity


M.C. Escher’s name also came up in the fractal/art discussion. Escher (1898-1972) was a Dutch graphic artist whose most well-known work was highly mathematical, incredibly complex, and, unbelievably, hand-made. Making many of his works with woodcuts, his technical expertise was staggering - this is not someone using a computer program, but instead pencil, paper, wood, knives, and ink.Many Escher prints suggest infinity , but in a somewhat different way from a basic fractal such as the Sierpinski triangle. The image at the top of this post - titled “Circle Limit III” - implies an infinity of interlocking fish, with the surface of the sphere as the “inhabitable” universe. The visualization of infinity is suggested by the diminishing views in perspective at the sphere boundary. While the creatures near the edge are self-similar to the middle-sphere images, the Sierpinski fractal’s self-similarity is found by zooming Into a section, rather than moving to the boundaries. Thus the Escher image is more akin to the Mandelbrot bug, appearing at different sizes and orientations along tendrils emanating from the main bug. (For the mathematics behind the graphic, see The Trigonometry of Escher’s Woodcut “Circle Limit III”, by H. S. M. Coxeter.)583047-430652-thumbnail.jpg

Ascending and DescendingEscher also encapsulated an infinity of time in some of his prints - most famously displayed in “Ascending and Descending,” a print that was one of his set of “impossible structures.” Are the figures in the picture walking up or down the stairs? And when do they reach the top (or bottom)?Note that the Escher web site contains a free interactive puzzle in which your task is to build an impossible structure.Read about more of the mathematics behind another of Escher’s famous works - The Print Gallery - as well as the connection with Dutch chocolate at the web site Escher and the Droste Effect. (Be sure to check out the animations at this site.) Here you’ll find ample mention of scaling - a necessary feature of fractals.