[Marxism] science, religion, islam (penrose tilings)
Les Schaffer
schaffer at optonline.net
Mon Feb 26 07:14:13 MST 2007
you need to see the pictures of these tilings too, i can send offlist to
interested readers.
Les
Science 23 February 2007:
Vol. 315. no. 5815, p. 1066
DOI: 10.1126/science.315.5815.1066
News of the Week
MATHEMATICS:
Quasi-Crystal Conundrum Opens a Tiling Can of Worms
John Bohannon
The mosques and palaces of the medieval Islamic world are wonders of
design. Because tradition forbids any pictorial decorations, they are
covered with complex and intricate mosaics. These geometric patterns,
called girih in Arabic, may be even more sophisticated than has been
appreciated.
Figure 1 Middle Age masters. The medieval architects who created
complex tiling patterns, such as these on a madrasa in Bukhara,
Uzbekistan, may have been more sophisticated than has been appreciated.
On page 1106, physicists Peter Lu of Harvard University and Paul
Steinhardt of Princeton University propose that architects made a
conceptual breakthrough sometime between the 13th and 15th centuries. By
first visualizing a surface as a tiling of polygons, these unknown
scholars created girih patterns that conform almost exactly to a pattern
called a quasi-crystal. If Lu and Steinhardt are right, then the Islamic
world discovered a piece of mathematics 500 years before it was formally
described in the West. But the paper has also sparked a rancorous
dispute over who first made this insight, and whether it is true at all.
Starting in the 1960s, mathematicians studying the geometry of tiling
came up with the concept of the quasi-crystal. Tiling is crystalline if
it is made up of an infinitely repeating pattern of some finite set of
units. Quasi-crystals are also made up of a finite set of interlocking
units, but their pattern never repeats even if tiled infinitely in all
directions. Researchers also found that although pentagons and decagons
don't fit easily into normal tiling, in a quasi-crystal such fivefold
and 10-fold rotational symmetries are integral. The most famous
quasi-crystal pattern is "Penrose tiling," named after Oxford University
mathematician and cosmologist Roger Penrose.
In 2005, Lu, a doctoral student at Harvard, noticed a geometric pattern
on the wall of an Islamic school in Uzbekistan with surprisingly complex
decagonally symmetric motifs. "It got me thinking that maybe
quasi-crystals had been discovered by Islamic architects long ago," he
says. Islamic architects began to explore motifs with fivefold and
10-fold rotational symmetry during a flourishing of geometric artistry
between the 11th and 16th centuries.
Back at Harvard, Lu began to study architectural scrolls from that
period. On many scrolls, faintly sketched beneath the intricate lines of
the girih design, was a polygonal tiling pattern. "I found the outlines
of the same tile shapes appearing over and over," he says. Lu realized
that Islamic architects could have used a pattern of polygonal
shapes--which he calls girih tiles--as the starting point for their
designs, creating a wonderfully complex girih pattern by tracing lines
from tile to tile following local rules. And if the right shape of girih
tiles were laid together just so, the resulting pattern could be
extended forever without repeating--a quasi-crystal.
Lu examined "a few thousand" photos of real mosques and found that
although decagonal girih patterns became increasingly common from 1200,
nearly all are periodic and so are not quasi-crystals. But then he found
a photo of the Darb-i Imam shrine in Isfahan, Iran, built in 1453. Its
decagonally symmetric motifs on two different length scales are a
telltale sign of a quasi-crystal. Working with Steinhardt, his former
undergraduate adviser and a quasi-crystal expert, Lu found that the
Darb-i Imam girih pattern can map onto a Penrose tiling. There were a
few defects, but these are superficial, says Lu, and were likely
introduced by workers during construction or repair. "We realized that
by the 15th century, these architects had the makings of
quasi-crystals," says Lu.
The paper has had a mixed reception. Crystal expert Emil Makovicky of
the University of Copenhagen, Denmark, studied girih patterns for 2
decades. His analysis of the patterns on a tomb in Maragha, Iran, built
in 1197, concluded that they map onto Penrose tiles and was published in
a 1992 book about fivefold symmetry. Lu and Steinhardt cite his work, he
says, but "without proper quoting and … in a way that [the ideas] look
like their own."
Physicist Dov Levine of the Israel Institute of Technology in Haifa
agrees that Makovicky deserves more credit than he is given in the
paper. "His analysis of [the Maragha tomb] patterns anticipates some of
the ideas in the Lu and Steinhardt paper," he says. Joshua Socolar, a
physicist at Duke University in Durham, North Carolina, agrees that
Makovicky deserves credit for discovering "an interesting relation
between the Maragha pattern and a Penrose tiling with a few defects."
Both Levine and Socolar doubt that the architects truly understood
quasi-crystals but say Lu and Steinhardt have generated interesting and
testable hypotheses.
Lu and Steinhardt say they were aware of Makovicky's published work on
the subject, but "we have found serious problems with both his technical
reconstruction and general conclusions." They say that they decided to
limit their references to Makovicky "to avoid having to address the
serious technical problems with his work." Makovicky disagrees that his
work is flawed.
Beyond the question of credit, just how mathematically sophisticated
these medieval architects really were remains open. "We haven't done an
exhaustive search of Islamic architecture by any means," says Lu. "There
could be a perfect quasi-crystal pattern waiting to be found."
REPORTS
Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture
Peter J. Lu and Paul J. Steinhardt (23 February 2007)
Science 315 (5815), 1106. [DOI: 10.1126/science.1135491]
=====
Science 23 February 2007:
Vol. 315. no. 5815, pp. 1106 - 1110
DOI: 10.1126/science.1135491
Reports
Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture
Peter J. Lu* and Paul J. Steinhardt
The conventional view holds that girih (geometric star-and-polygon, or
strapwork) patterns in medieval Islamic architecture were conceived by
their designers as a network of zigzagging lines, where the lines were
drafted directly with a straightedge and a compass. We show that by 1200
C.E. a conceptual breakthrough occurred in which girih patterns were
reconceived as tessellations of a special set of equilateral polygons
("girih tiles") decorated with lines. These tiles enabled the creation
of increasingly complex periodic girih patterns, and 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.
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