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A monthly survey of math news
This month's topics:
Rudresh Mahanthappa makes music from Fibonacci numbers. Illustration by Joe Sutliff, from Science 315, p. 462, used with permission.
"Riffs on Numerical Themes" is a piece in the January 26 2007 Science. John Bohannon takes us with him to a "sort of high-end jazz bar" in Greenwich Village, to hear the saxophonist Rudresh Mahanthappa and his quartet. He has brought along Michael Thaddeus, an algebraic geometer at Columbia, because Mahanthappa is performing pieces from his CD "Codebook" composed using number theory. In "Further and Beyond," for example: "The melodies batted between sax, bass and piano are permutations of a scale built on the semitone intervals (1, 4, 2, 8, 5, 7)," Bohannon tells us, and explains that 142857 is a "cyclical" number with the property that any product with an integer between 1 and 6 permutes its digits. In "The Decider," another example, "The manic melodies of the tune ... are a mapping of the Fibonacci sequence onto the 12-tone musical scale." (Thaddeus: "The mathematical themes are difficult to hear.") Bonahon asks Manhanthappa if this is just a gimmick. No. The constraint itself enters into the creative process, and furthermore the Fibonacci sequence is unique. "It sounds right no matter what key the others are comping [accompanying] in. I tried alternative sequences and they don't have that property." Bohannon meditates on "the heightening of the senses from an awareness of hidden layers of meaning," etc., but does include this wonderful quote from Leibniz: "Music is the pleasure the human mind experiences from counting without being aware that it is counting."
A quasi-periodic tiling from the Darb-i Imam shrine in Isfahan. Image courtesy K. Dudley and M. Elliff.
"... by the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West." This text appears in the abstract for "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture," by Peter J. Lu and Paul J. Steinhardt, in the February 23 2007 Science. It is known that 5-fold rotational symmetry is incompatible with translational periodicity, but Peter Lu seems to have been the first one to notice that medieval Islamic artists went ahead, used motifs with 5-fold symmetry, and produced "quasi-periodic" patterns long before that concept was born. As he told NPR (All Things Considered, February 22 2007; transcript and images available online), he made this observation during a trip to Uzbekistan. When he got back to Harvard, where he is a graduate student in Physics, he did some investigation and discovered that Islamic geometers had devised a set of five polygonal building-blocks, each one decorated with polygonal lines; when the blocks were used to tile an area the lines fit together to give the intricate knot-like patterns called girih. One of these "girih blocks" is in fact identical to the "fat rhombus" we use in Penrose tilings.
Part of a girih-pattern tiling from a Turkish mosque, with its analysis in terms of three of the decorated blocks (bowtie, decagon and flat hexagon) used by Islamic geometers. The other two are a pentagon and our "fat rhombus." Photographic image courtesy W. B. Denny, geometric analysis image courtesy Peter J. Lu.
Self-similarity is the one of the hallmarks of Penrose-type quasi-priodic tilings; this fact also seems not to have escaped the Islamic geometers: "Perhaps the most striking innovation arising from the application of girih tiles was the use of self-similarity transformation (the subdivision of large girih tiles into smaller ones) to create overlapping patterns at two different length scales, in which each pattern is generated by the same girih tile shapes." An example: the tiling from the Darb-i Imam shrine shown above.