Penrose tiling Down Under
Three earlier examples of Penrose-tiled floors: from top, Carleton College, Feldman Residence, Meredith College. Images used with permission; use of Meredith College photograph courtesy of Sir Roger Penrose.
Martin Kemp has a "Science and Culture" piece in the July 21 2005 Nature, entitled "A trick of the tiles." The subject is the floor of the atrium of the new Molecular and Chemical Sciences Building at the University of Western Australia in Perth. Using "two types of locally manufactured concrete tile in the form of fat and thin rhombuses," the floor was made into an aperiodic Penrose tiling. This is not the first published Penrose tiling of a floor. Predecessors include the 1979 terazzo courtyard of Bachelor Hall at Miami University, the 1993 ceramic cartwheel tiling at Carleton College, the 1997-98 ceramic bathroom floor chez Alex Feldman (a math professor at Boise State University) and the 2003 terazzo atrium of the Science and Mathematics Building at Meredith College (Raleigh, NC). What distinguishes the Perth tiling is the material (concrete) and the use of the 2-rhombus form of the Penrose tiling instead of the dart-and-kite form used in the earlier examples. The two tilings have the same mathematical properties, but they affect the eye differently. Kemp describes what happens when we look at a 2-rhombus tiling:
A Penrose tiling like that used in Perth. Each tile is a rhombus with smaller angle 72o ("fat") or 36o ("thin"). They are joined using matching rules which force aperiodicity. Image generated by Stephen Collins' program Bob.
"Almost inevitably, spatial instincts ... come into play ... . We can, for instance, play Necker cube-type games with apparent octagons, and facet the surface into a kind of cubist medley of receding and advancing planes." Kemp anchors this point in Art History by showing some surprisingly pre-Penrosesque doodles by Albrecht Dürer, dated 1524. Math and the Unicorn Tapestries
Richard Preston's "Capturing the Unicorn," subtitled "How two mathematicians came to the aid of the Met" appeared under the Art and Science rubric in the New Yorker for April 11, 2005. It turns out that the Metropolitan Museum had a problem. The Unicorn Tapestries, the crown jewels of the Met's Medieval collection, were taken down for cleaning in 1998, and were photographed then as part of the Museum's high-resolution digital record project. The tapestries -there are six of them and a fragmentary seventh- are typically twelve feet high and somewhat wider. The digital Leica set up to do the job could only capture one 3 by 3-foot square at a time. But assembling the digital files into a coherent image was too large a job for for the Museum's computers to handle. The data -more than two hundred CDs- were filed away, and the tapestries reinstalled on the museum walls. Fast forward to 2003, when David Chudnovsky meets a Metropolitan curator at a dinner party. He and his brother Gregory ("The Chudnovsky brothers claim they are one mathematician who happens to occupy two human bodies") soon take on the computing job, which should be a snap for their latest home-built supercomputer (called "the Home Depot thing" or just "It"). But there is a twist: even after geometric transformations have corrected for all possible perspective changes between adjacent frames, the images on the overlaps are hopelessly out of registration: it's as if the tapestry were a living being which had taken a breath between takes. Everything has slightly shifted. Coaxing the overlaps back into registration requires a new "warping" technique, as the Chudnovskys explain it, a 2-dimensional analogue of techniques used in DNA sequencing and speech recognition. The computation is huge: it takes the "Chudnovsky Mathematician" -Preston's coinage- and "It" three months to process just one tapestry, but "The Unicorn in Captivity" is digitally captured in seamless splendor.
The left-hand side shows the overlap between two adjacent frames of the photo mosaic, after all perspective corrections have been made. The right-hand side shows the two images brought into registration by the Chudnovskys' warping technique. Images courtesy Tom Morgan, IMAS.
End of story: One tapestry, apparently, was enough. The brothers have moved on to a bigger project, working on the design of what may be the world's most powerful supercomputer, for "a United States government agency."
Stony Brook University
tony at math.sunysb.edu