neverendingbooks

According
to a science article in the New York Times, archeologists have discovered “signs of advanced math” in
medieval mosaics. An example of a quasi-crystalline Penrose pattern was
found at the Darb-i Imam shrine in Isfahan, Iran.

A new study shows that the Islamic pattern-making process, far more
intricate than the laying of one’s bathroom floor, appears to have
involved an advanced math of quasi crystals, which was not understood by
modern scientists until three decades ago. Two years ago, Peter J. Lu,
a doctoral student in physics at Harvard University, was transfixed by
the geometric pattern on a wall in Uzbekistan. It reminded him of what
mathematicians call quasi-crystalline designs. These were demonstrated
in the early 1970s by Roger Penrose, a mathematician and cosmologist at
the University of Oxford. Mr. Lu set about examining pictures of other
tile mosaics from Afghanistan, Iran, Iraq and Turkey, working with Paul
J. Steinhardt, a Princeton cosmologist who is an authority on quasi
crystals and had been Mr. Lu’s undergraduate adviser.

Penrose tilings are one of the motivating examples of Alain Connes’
book as there is a
$C^* $-algebra associated to it. In fact, the algebra is AF ( a limit of
semi-simple finite dimensional algebras) so is even a formally smooth
algebra in Kontsevichian noncommutative geometry (it is remarkable how
quickly one gets used to silly terminology…). However, the Penrose
algebra is simple, so rather useless from the point of view of finite
dimensional representations… Still, Connesian noncommutative geometry
may be a recent incarnation of the medieval Tehran program (pun
intended). Thanks to
easwaran for the link
(via Technorati).

_Added, March 1_ : I haven’t looked at the
Connes-Marcolli paper A walk in the noncommutative garden for a while but now
that I do, I see that they mentioned the above already at the end of
their section on Tilings (page 32). They also include clearer pictures.