Also available at http://math.ucr.edu/home/baez/week281.html
October 19, 2009
This Week's Finds in Mathematical Physics (Week 281)
John Baez
This week I'd like to finish my news report from the Corfu summer
school on quantum gravity. You'll hear how strings meet loops in BF
theory, and how the Poincare 2-group gives a spin foam model that
mimics flat Minkowski spacetime.
But first: Timurid tiling patterns with 5-fold and 10-fold
quasisymmetry, and the astronomy pictures of the week!
If you listen to the news, you probably heard that NASA discovered an
enormous diffuse ring around Saturn. They did it using the Spitzer
Space Telescope, a satellite equipped with a telescope that detects
infrared light. In "week243" I showed you infrared light from the
first stars in the Universe, and in "week257" I talked about magnesium
and iron oxide dust emanating from the Red Rectangle. Both of those
were discovered using the Spitzer.
Here's what the new ring would look like if you could see it:
1) Jet Propulsion Laboratory, NASA space telescope discovers largest
ring around Saturn, October 6, 2009,
http://www.jpl.nasa.gov/news/news.cfm?release=2009-150
As you probably heard, it dwarfs all the visible rings, and it's
tilted relative to them. But even cooler is what the Spitzer Space
Telescope actually saw:
2) NASA, Big band of dust,
http://www.nasa.gov/mission_pages/spitzer/multimedia/spitzer-20091007...
It's an edge-on view of the new ring. It's fat: 20 Saturns thick.
And if you look carefully, you'll see that it has two layers, with a
bit of a gap in the middle. According to the scientists who
discovered it, this is consistent with its origin:
3) Anne Verbiscer, Michael Skrutskie, and Doug Hamilton, Saturn's
largest ring, Nature, published online October 7, 2009.
The point is that this ring surrounds the orbit of Saturn's moon
Phoebe - a meteor-scarred hulk 100 kilometers across. While Phoebe
looks like an asteroid, it's probably an interloper from the outer
Solar System, because it's made of ice. But it's covered with a
layer of dark material.
The newly discovered ring seems to be made of this dark stuff, blasted
away from Phoebe by meteorite collisions. And its discoverers say the
double-layered structure is characteristic of rings formed this way
from moons with inclined orbits. (Jupiter also has some faint rings
like this, poetically known as the "gossamer rings".)
What's really exciting about this new ring is that it explains one of
the big mysteries of the Solar System: the dark spot on Saturn's moon
Iapetus!
Iapetus is mostly icy, but one side is covered with dark stuff...
probably cyanides and carbon-rich minerals. Now it seems this stuff
was picked up from the newly discovered ring! It seems to have landed
in lumps - mainly on the leading side of Iapetus. You see, this moon
is locked in synchronous rotation with Saturn, so the front plows
through space and picks up debris, while the back stays clean.
Here's a closeup of some lumps of dark stuff on Iapetus, taken by the
Cassini probe:
4) NASA, Inky stains on a frozen moon, September 12, 2007,
http://saturn.jpl.nasa.gov/photos/imagedetails/index.cfm?imageId=2733
If you want more, there's a great introduction to Saturn's rings
in this blog, followed by a nontechnical summary of the new paper
on the Phoebe ring:
5) Emily Lakdawalla, The Phoebe ring, The Planetary Society Blog,
October 14, 2009, http://planetary.org/blog/article/00002165/
As Lakdawalla points out, discovering a big ring was just the
beginning:
So far, it's a cool result but it's sort of like stamp collecting -
we discovered a new X and described it, done. Where the paper gets
really interesting is when the authors explore what happens to the
particles in Phoebe's ring over time, something that you can model
by writing down a few equations that describe the orbit of a
particle, include Saturn, Phoebe, Iapetus, and Titan, include
the masses, densities, and albedos of the particles, and the
effects of incident sunlight.
What happens to particles depends upon their size. The biggest
chunks, several centimeters in size or larger, don't really
migrate anywhere, sticking around near Phoebe's orbit until
they smack into something - each other or Phoebe. The model
simulation suggests that it would take more than the age of
the solar system for half of the particles to be removed from
the system by re-collision with Phoebe, so most of the biggest
chunks are still out there somewhere in Phoebe's orbital space.
What about smaller particles? The article says "re-radiation
of absorbed sunlight exerts an asymmetric force on dust grains,
causing them to spiral in towards Saturn with a characteristic
timescale of 1.5 x 10^5 r years, where r is the particle radius in
micrometers. This force brings all centimetre-sized and smaller
material to Iapetus and Titan unless mutual particle collisions
occur first.... Most material from 10 micrometres to centimetres
in size ultimately hits Iapetus, with smaller percentages striking
Hyperion and Titan." This would be a slow process that has
operated continuously since whenever Phoebe was captured into
Saturn's orbit. There might have been bursts of material delivered
to Iapetus associated with some of the bigger impacts that have
left such large scars on Phoebe, but they would have been blips
above a steady background.
Next: tilings. The science fiction writer Greg Egan is also a
professional programmer, and he's written a remarkable collection
of Java applets, which you can see on his website. Here's the latest:
6) Greg Egan, Girih, http://www.gregegan.net/APPLETS/32/32.html
This program generates quasiperiodic tilings with approximate 10-fold
rotational symmetry using a method called "inflation". The idea of
inflation is to take a collection of tiles and repeatedly subdivide
each one into smaller tiles from the same collection. Egan's applet
shows the process of inflation at work: patterns zooming in endlessly!
Some of the math behind this is modern, but some goes back to the
Timurids: the dynasty founded by the famous conqueror Timur, also
known as Tamerlane. By 1400, the Timurid empire was huge. It
included most of central Asia, Iran, and Afghanistan, as well as
large parts of Pakistan, India, Mesopotamia and the Caucasus. Its
capital was the magical city of Samarkand.
The Timurids raised the art of tiling to its highest peak. Islamic
artists had already explored periodic tilings with most of the 17
mathematically possible "wallpaper groups" as symmetries - for more
on this, see my tour of the Alhambra in "week267". What was left
to do? Well, periodic tilings can have 2-fold, 3-fold, 4-fold,
or 6-fold rotational symmetry, but nothing else. Notice the gap?
It's the number 5! So that's what they tackled.
Precisely because you *can't* produce periodic tilings with 5-fold
rotational symmetry, it's a delightful artistic challenge to fool
the careless eye into thinking you've done just that.
In the 1970's, Penrose discovered quasiperiodic patterns with
approximate 5-fold symmetry - for example, patterns made of two
tiles, called "kites" and "darts":
* kite: a convex quadrilateral with interior angles of 2pi/5,
2pi/5, 2pi/5 and 4pi/5 as you march around it.
* dart: a nonconvex quadrilateral with interior angles of 2pi/5,
pi/5, 6pi/5 and pi/5.
The work of Penrose launched a huge investigation into quasiperiodic
tilings and quasicrystals. With their eyes opened, modern scientists
saw how fascinating the old Timurid tilings were:
7) Peter J. Lu and Paul J. Steinhardt, Decagonal and quasi-crystalline
tilings in medieval Islamic architecture, Science 315 (2007), 1106–1110.
Also available at
http://www.physics.harvard.edu/~plu/publications/Science_315_1106_200...
Lu and Steinhardt described a set of 5 tiles which seem to underlie a
lot of Timurid designs:
* a regular pentagon with five interior angles of 3pi/5.
* a regular decagon with ten interior angles of 4pi/5.
* a rhombus with interior angles of 2pi/5, 3pi/5, 2pi/5, 3pi/5.
* an elongated hexagon with interior angles of 2pi/5, 4pi/5,
4pi/5, 2pi/5, 4pi/5, 4pi/5.
* a bow tie (non-convex hexagon) with interior angles of
2pi/5, 2pi/5, 6pi/5, 2pi/5, 2pi/5, 6pi/5.
All the edges of all these tiles have the same length.
Lu and Steinhardt call these "girih tiles". But "girih" actually
means "strapwork": the braided bands that decorate the tiles in a lot
of this art. Egan's applet uses three of these tiles: the decagon,
the elongated hexagon and the bowtie. As you'll see on his webpage,
each can be subdivided into smaller decagons, hexagons and bowties.
And that's how "inflation" works.
Did the Timurid artists actually understand the process of inflation,
or the idea of a quasiperiodic tiling? Seeking clues, scholars have
turned to the Topkapi Scroll, a kind of "how-to manual" for tiling
that resides in the Topkapi Palace in Istanbul. I would love to get
my paws on this color reproduction:
8) Gulru Necipoglu and Mohammad al-Asad, The Topkapi Scroll - Geometry
and Ornament in Islamic Architecture, Getty Publications, 1996.
I've put it on my wishlist at Amazon - but a used copy costs almost
$2000, so I'm not holding my breath. For now, the best substitute
I've found is this beautiful article:
9) Peter R. Cromwell, The search for quasi-periodicity in Islamic
5-fold ornament, Math. Intelligencer 31 (2009), 36-56.
Also available at
http://www.springerlink.com/content/760261153n347478/?p=405b9dbf45ea4...
The Mathematical Intelligencer is a wonderful magazine put out by
Springer Verlag. It's recently become available online - and to my
shock the above article is free! Springer doesn't give much away,
so I can't help but fear this is an oversight on their part, soon to
be corrected. So,
...
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