Thursday, March 1, 2007

in medieval architecture, signs of noncommutative geometry?

MOSAIC SOPHISTICATION A quasi-crystalline Penrose pattern at the Darb-i Imam shrine in Isfahan, Iran

A few days ago I noticed this article in NYT science section that reports on a recent paper by Lu and Steinhardt in Science (see here and here for the full article; thanks to `thomas1111'). Their abstract says: ``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''.

Interestingly enough the occurrence of quasi periodic tilings in old Persian art was also extensively commented on, last year, in Alain and Matilde's article ``A walk in the noncommutative garden" (see Section 9 on tilings). The first four pics are from their article. (see also lieven le bruyn’s weblog where the NYT article is commented at). We look forward to comments by people in NCG, operator algebras, and those working on quasi periodic crystals.


Anonymous said...

This paper by Lu and Steinhardt is apparently available on Lu's website here
an additionnal material here.

A related question from a beginner: is there a relation between primes or Gaussian primes and quasi-crystals (i.e. can their respective aperiodicity be somehow related) ?

Anonymous said...

Dear Thomas1111,
Thanks a lot for pointing to the exact address for the paper by Lu and Steinhardt. Your question is very relevant and I think the answer is yes, in the following sense. One of the features of Connes' program for Riemann hypothesis (which is after all about the distribution of primes) is that to understand the set of primes one is `forced' to do geometry, topology, and analysis on a highly singular space, in fact a noncommutative space. Now this noncommutative space is more complicated that the space
of quasi-crystals, but I guess there are some links. May be Alain will find time to shed light on this issue?

Mahndisa S. Rigmaiden said...

03 08/ 07

I love the way non commutivity pops up in art from various cultures. Certain African hairstyles reveal tesselation patterns for a hexagonal lattice! And I also read about some Indian artistry that women perform before they are married which has wonderful fractal characteristics.

I guess the point is that there are few new things under the sun, but how we interpret those things is richly variant. Nice post.