Tony Phillips' Take on Math in the Media A monthly survey of math news

This month's topics:

• Antikythera link to Archimedes?
• Archimedean approximation to Penrose tiling
• "Swinging with math"

Antikythera link to Archimedes?

The Antikythera Mechanism is back in the news. As reported in the July 31 2008 Nature, new observations, using "microfocus X-ray computed tomography (CT)" have allowed the discovery and deciphering of previously unknown or misunderstood dials and inscriptions on the mechanism. The report (team led by Tony Freeth of the Antikythera Mechanism Research Project, Cardiff) emphasizes the following details:

• "The upper back dial is a 19-year calendar, based on the Metonic cycle, arranged as a five-turn spiral." The names of the months (all 12 are identified) are "unexpectedly of Corinthian origin." Since the principal Corinthian settlement extant at the time of the Mechanism (near 100 BC) was Syracuse, this suggests "a possible mechanical tradition going back to Archimedes (died 212 BC), who invented a planetarium described by Cicero (first century BC) and wrote a lost book on astronomical mechanisms."
• "The upper subsidiary dial is not a 76-year Callippic dial as previously thought, but follows the four-year cycle of the Olympiad and its associated Panhellenic Games." The authors conclude that "the Antikythera Mechanism was not simply an instrument of abstract science, but exhibited astronomical phenomena in relation to Greek social institutions."

This story, coming so shortly before the 2008 Summer Olympics, was widely picked up by the media. Dan Vergano had a piece in the July 31 USA Today: "Antikythera Mechanism helped Greeks set Olympic schedule"; John Noble Wilford had "Discovering How Greeks Computed in 100 B.C." in that day's New York Times; The New Scientist ran "Ancient Greek computer could have roots in Archimedes' ideas" on August 2.

Archimedean approximation to Penrose tiling

"Archimedean-like tiling on decagonal quasicrystalline surfaces" appeared in Nature for July 24, 2008. A Stuttgart team led by Clemens Bechinger devised an experimental apparatus that morphs a uniform triangulation of the plane into a Penrose tiling with, as an intermediate step, an Archimedean-like tiling with rows of squares and equilateral triangles.

Configuration of the monolayer as a function of the charge intensity on the quasi-crystalline substrate relative to the electrostatic repulsion between the particles in the layer. a low, c intermediate, e high. Initially the particles organize themselves into an almost pure triangular lattice, oriented along one of the basis directions of the substrate. When the intensity is high, the particles replicate the quasi-crystalline structure. But for an intermediate intensity, the particles approximate an archimedean tiling of the plane by squares and equilateral triangles. The images b, d, f show diffraction patterns from the monolayer, with the arrows indicating diffraction peaks also found in f. Image from Nature 454 501-504, used with permission.

The quasicristalline substrate was realized as the complex of optical gradients associated to the interference pattern between 5 parallel laser beams. The monolayer was formed by a colloidal suspension of highly charged polystyrene spheres of radius R = 1.45 μm. When the laser beams were turned off, electrostatic repulsion between the beads resolved them into a triangular-lattice "crystalline" configuration. When the laser intensity was raised, and the inter-particle repulsion was reduced, the gradients guided the beads into a quasi-crystalline configuration duplicating a Penrose tiling, and giving a diffraction pattern with 10-fold rotational symmetry. What is surprising is that there exists an intermediate configuration closely related to another recognizable regular structure: the planar tiling by squares and equilateral triangles, one of the "Archimedean" tilings listed by Kepler in 1619. The authors show how this elementary tiling approximates significant features of its non-periodic cousin.

The square-triangle Archimedean tiling has quasi-pentagonal symmetries (the angle γ = 75^o is close to π/5 = 72^o.)

"Swinging with math"

This is the title of a short item in Science for August 1, 2008. The "swinger" is golfing champion Phil Mickelson, who has made an ad for the Mickelson ExxonMobil Teachers Academy in which he "tees off as equations dance in the foreground." As he explained to the House Education and Labor Committee: "I use statistics to maximize my practice. I do a drill with 3-foot putts. And I can make 100% of them. But at 4 feet it's 88% and ... at 6 feet it's only 65%. So ... what I really need to do is hit my chip shots within 3 feet of the hole." The Academy trains elementary-school teachers (1400 so far) in 1-week summer sessions.

Tony Phillips
Stony Brook University
tony at math.sunysb.edu

Recent Math Digest Summaries:
Posted here September 2008

"Educators Peer Over Students' Shoulders at Mich. Math Lab," by Sean Cavanagh. Education Week, 11 August 2008.

New teachers are sometimes encouraged or required to attend a master teacher's class and observe. But a University of Michigan program gives teachers and those studying to become teachers a unique opportunity to immediately follow up their observation of the class with access to copies of student work, discussion with peers and education researchers, and questions to the master teacher. This program, says teacher and researcher Deborah Ball, seeks to answer a question, "What's the math you need to know to teach?" and in doing so "expose the myth people have that teaching elementary math is easy."

This model of a "laboratory/classroom" allows educators to test teaching techniques, develop curricula, and help teachers analyze teaching styles by watching them in action rather than reading about them in a book or hearing about them in a lecture. Classes are videotaped and student work is photocopied for future reference. The idea of a model open classroom is not new, and originated in Japan as a research tool. But the inclusion of both academics and in-service teachers in the discussion and analysis of model lessons may help bridge the gap between mathematicians' and classroom-teachers' visions of elementary education.

--- Brie Finegold

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Reviews Books, plays and films about mathematics

Citations for reviews of books, plays, movies and television shows that are related to mathematics (but are not aimed solely at the professional mathematician). The alphabetical list includes links to the sources of reviews posted online, and covers reviews published in magazines, science journals and newspapers since 1996.

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