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-20091007d.html
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, just like our Moon
always shows the same face to Earth. So, one side plows through space
and picks up debris, while the other stays clean.
Here's a closeup of some lumps of dark stuff on Iapetus, taken by the
Cassini probe:
4) NASA Photojournal, Spotty Iapetus,
http://photojournal.jpl.nasa.gov/catalog/PIA08382
NASA Photojournal, Inky stains on a frozen moon,
http://photojournal.jpl.nasa.gov/catalog/PIA08374
As you can see, in this region of Iapetus the dark stuff is found at
the bottoms of craters. It could have formed these craters by impact,
but its presence could also gradually make these craters deeper: the
dark stuff should absors more sunlight and warm the nearby ice, making
it "sublimate": that is, turn into water vapor.
Because it's locked in synchronous rotation with Saturn, the "day" on
Iapetus is equal to one period of rotation, namely 79 of our Earth
days. So, it's probably the warmest place in the Saturnian system
during the daytime. Not very warm: just 113 kelvin on the ice.
That's -160 degrees Celsius! But in the dark regions it should be
about 138 kelvin. This extra warmth should make more ice sublimate,
making them even darker. It's been estimated that over one billion
years the very dark regions would lose about 20 meters of ice to
sublimation, while the light-colored regions would lose only 10
centimeters, not even counting the ice transferred from the dark
regions.
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.
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.
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=405b9dbf45ea4f4793a097b6e12dcb08pi=7
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, grab a copy of this article *now*.
Cromwell argues that we shouldn't attribute too much modern mathematical
knowledge to the Timurid tile artists. But the really great thing about
this article is the detailed information on how some of these tiling
patterns are made - including lots of pictures. It repays repeated study.
Here's a less mathematical and more historical introduction to the
Timurid tile artists, also with lots of nice pictures:
10) Sebastian R. Prange, The tiles of infinity, Saudi Aramco World
(October-November 2009), 24-31. Also available at
http://www.saudiaramcoworld.com/issue/200905/the.tiles.of.infinity.htm
You should also check out Craig Kaplan's work. He's studied Kepler's
work on patterns built from decagons, and written software that
generates beautiful star patterns:
11) Craig Kaplan, The trouble with five, Plus Magazine 45 (December
2007), available at
http://plus.maths.org/issue45/features/kaplan/
12) Craig Kaplan, A meditation on Kepler's Aa, in Bridges 2006:
Mathematical Connections in Art, Music and Science, 2006, pp. 465-472.
Also available at http://plus.maths.org/issue45/features/kaplan/
13) Craig Kaplan, Taprats: computer generated Islamic star
patterns, http://www.cgl.uwaterloo.ca/~csk/washington/taprats/
Together with David Salesin, he's also gone beyond the old masters by
studying tilings in spherical and hyperbolic geometry:
14) Craig S. Kaplan and David H. Salesin, Islamic star patterns in
absolute geometry, ACM Transactions on Graphics 23 (April 2004),
97-119. Also available at
http://www.cgl.uwaterloo.ca/~csk/papers/tog2004.html
Another key player in this business is Eric Broug:
15) Broug Ateliers: Islamic Geometric Design, http://www.broug.com/
Check out the nice image galleries! I bought this book, which
explains how to make these patterns yourself:
16) Eric Broug, Islamic Geometric Patterns (book with CD-ROM), Thames
and Hudson, 2008.
I'll list a bunch more references below, for when I retire and get
time to devote myself more deeply to this subject. But now - on to
Corfu!
Last time I said a bit about what I learned in Ashtekar and Rovelli's
courses. Now I'd like to talk about some other things I learned in
Corfu - some things I find even more tantalizing.
In "week232", I explained how gravity in 3d spacetime automatically
contains within it a theory of point particles, and how a 4d analogue
of 3d gravity automatically contains within it a theory of string-like
objects. This 4d theory is called BF theory. Like 3d gravity, it
describes a world where spacetime is flat. So, it's boring compared
to full-fledged 4d gravity - so boring that we can understand it much
better! In particular, unlike 4d gravity, we understand a lot about
what happens when you take quantum mechanics into account in 4d BF
theory.
But when you remove a surface from spacetime in 4d BF theory, it
springs to life! In particular, the surface acts a bit like the
worldsheet of a string. It doesn't behave like the strings in
ordinary string theory: its action is not equal to its area. But
Winston Fairbairn has been thinking about this a lot:
17) Winston J. Fairbairn and Alejandro Perez, Extended matter coupled
to BF theory, Phys. Rev. D78:024013, 2008. Also available as
arXiv:0709.4235.
18) Winston J. Fairbairn, On gravitational defects, particles and
strings, JHEP 0809:126, 2008. Also available as arXiv:0807.3188.
19) Winston J. Fairbairn, Karim Noui and Francesco Sardelli, Canonical
analysis of algebraic string actions, available as arXiv:0908.0953.
And it turns out that if we impose the constraints on BF theory that
turn it into general relativity, we obtain the usual Nambu-Goto
string, where the action is the area! However, the last of the three
papers above shows there are some subtle differences.
I need to think about this a lot more. It was always my hope to
reconcile string theory and loop quantum gravity, and this could be
the way. Of course, reconciling two things that don't work doesn't
necessarily give one that does. A pessimist might say that combining
string theory and loop quantum gravity is like combining epicycles and
aether. But I'm optimistic. Something interesting is going on here.
In a different but possibly related direction, Aristide Baratin gave
a talk on recent work he's been doing with Derek Wise and Laurent
Freidel. You can get a feel for this work from this paper:
20) Aristide Baratin, Derek K. Wise, 2-Group representations for spin
foams, to appear in proceedings of the XXV Max Born Symposium:
The Planck Scale, Wroclaw, Poland. Also available as
arXiv:0910.1542.
In "week235" I mentioned an amazing paper by Baratin and Freidel
called "Hidden quantum gravity in 4d Feynman diagrams: emergence of
spin foams". They described a spin foam model that acts just like
4-dimensional flat Minkowski spacetime: couple it to interacting
point particles, and you get the usual Feynman diagrams described
in a new way!
The big news is that this spin foam model comes from the
representations of a 2-group, instead of a group. Namely, the
Poincare 2-group. This is a 2-group I invented which has Lorentz
transformations as objects and translations as endomorphisms of any
object.
The Poincare 2-group spin foam model was first studied by Crane,
Sheppeard and Yetter. Baratin, Freidel, Wise and I spent a long time
developing the theory of infinite-dimensional representations of
2-groups needed to make this model precise - see "week274" for more on
all this. Now the details are falling into place, and a beautiful
picture is emerging.
I should admit that the paper by Baratin and Wise deals with the
Euclidean rather the Lorentzian version of this picture. I hope this
is merely because the representation theory of the "Euclidean 2-group"
is more tractable than that of the Poincare 2-group. I hope everything
generalizes to the Lorentzian case.
A lot to think about.
To wrap up, here's a big list of references from Cromwell's paper on
tilings I hadn't known so much had been written about this subject!
21) M. Arik and M. Sancak, Turkish-Islamic art and Penrose
tilings, Balkan Physics Letters 15 (1 Jul 2007) 1-12.
22) J. Bonner, Three traditions of self-similarity in fourteenth
and fifteenth century Islamic geometric ornament, Proc.
ISAMA/Bridges: Mathematical Connections in Art, Music and Science,
(Granada, 2003), eds. R. Sarhangi and N. Friedman, 2003, pp. 1-12.
23) J. Bonner, Islamic Geometric Patterns: Their Historical Development
and Traditional Methods of Derivation, unpublished manuscript.
24) J. Bourgoin, Les Elements de l'Art Arabe: Le Trait des Entrelacs,
Firmin-Didot, 1879. Plates reprinted in Arabic Geometric Pattern and
Design, Dover Publications, 1973.
25) J.-M. Castira, Arabesques: Art Decoratif au Maroc, ACR Edition,
1996.
26) J.-M. Castira, Zellijs, muqarnas and quasicrystals, Proc. ISAMA,
(San Sebastian, 1999), eds. N. Friedman and J. Barrallo, 1999, pp.
99-104.
27) G. M. Fleurent, Pentagon and decagon designs in Islamic art,
Fivefold Symmetry, ed. I. Hargittai, World Scientific, 1992, pp.
263-281.
28) B. Grunbaum and G. C. Shephard, Tilings and Patterns, W. H.
Freeman, 1987.
29) E. H. Hankin, On some discoveries of the methods of design employed
in Mohammedan art, J. Society of Arts 53 (1905) 461-477.
30) E. H. Hankin, The Drawing of Geometric Patterns in Saracenic Art,
Memoirs of the Archaeological Society of India, no 15, Government of
India, 1925.
31) E. H. Hankin, Examples of methods of drawing geometrical arabesque
patterns, Math. Gazette 12 (1925), 370-373.
32) E. H. Hankin, Some difficult Saracenic designs II, Math. Gazette
18 (1934), 165-168.
33) E. H. Hankin, Some difficult Saracenic designs III, Math. Gazette
20 (1936), 318-319.
34) A. J. Lee, Islamic star patterns, Muqarnas IV: An Annual on
Islamic Art and Architecture, ed. O. Grabar, Leiden, 1987, pp. 182.197.
35) P. J. Lu and P. J. Steinhardt, Response to Comment on "Decagonal
and quasi-crystalline tilings in medieval Islamic architecture",
Science 318 (30 Nov 2007), 1383.
36). F. Lunnon and P. Pleasants, Quasicrystallographic tilings, J.
Math. Pures et Appliques 66 (1987), 217-263.
37) E. Makovicky, 800-year old pentagonal tiling from Maragha, Iran,
and the new varieties of aperiodic tiling it inspired, Fivefold
Symmetry, ed. I. Hargittai, World Scientific, 1992, pp. 67-86.
38) E. Makovicky, Comment on "Decagonal and quasi-crystalline tilings
in medieval Islamic architecture", Science 318 (30 Nov 2007), 1383.
39) E. Makovicky and P. Fenoll Hach-Alm, Mirador de Lindaraja: Islamic
ornamental patterns based on quasi-periodic octagonal lattices
in Alhambra, Granada, and Alcazar, Sevilla, Spain, Boletin Sociedad
Espanola Mineralogia 19 (1996), 1-26.
40) E. Makovicky and P. Fenoll Hach-Alm, The stalactite dome of the
Sala de Dos Hermanas - an octagonal tiling?, Boletin Sociedad Espanola
Mineralogia 24 (2001), 1-21.
41) E. Makovicky, F. Rull Pirez and P. Fenoll Hach-Alm, Decagonal
patterns in the Islamic ornamental art of Spain and Morocco, Boletmn
Sociedad Espanola Mineralogia 21 (1998), 107-127.
42) J. Rigby, A Turkish interlacing pattern and the golden ratio,
Mathematics in School 34 no 1 (2005), 16-24.
43) J. Rigby, Creating Penrose-type Islamic interlacing patterns,
Proc. Bridges: Mathematical Connections in Art, Music and Science,
(London, 2006), eds. R. Sarhangi and J. Sharp, 2006, pp. 41-48.
44) F. Rull Pirez, La nocion de cuasi-cristal a traves de los mosaicos
arabes, Boletin Sociedad Espanola Mineralogia 10 (1987), 291-298.
45) P. W. Saltzman, Quasi-periodicity in Islamic ornamental design,
Nexus VII: Architecture and Mathematics, ed. K. Williams, 2008, pp.
153-168.
46) M. Senechal, Quasicrystals and Geometry, Cambridge Univ. Press,
1995.
47) M. Senechal and J. Taylor, Quasicrystals: The view from Les
Houches, Math. Intelligencer 12 (1990) 54-64.
Reference 24, the book by Bourgoin, is a classic - and the Dover version
is probably quite affordable. Cromwell also lists some more websites:
48) ArchNet. Library of digital images of Islamic architecture,
http://archnet.org/library/images/
49) E. Harriss and D. Frettlvh, Tilings Encyclopedia,
http://tilings.math.uni-bielefeld.de/
50) P. J. Lu and P. J. Steinhardt, Decagonal and quasi-crystalline
tilings in medieval Islamic architecture, supporting online material,
http://www.sciencemag.org/cgi/content/full/315/5815/1106/DC1
51) D. Wade, Pattern in Islamic Art: The Wade Photo-Archive,
http://www.patterninislamicart.com/
The last one is a huge treasure trove of images!
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Quote of the Week:
The arabesques displayed a profound use of mathematical principles, and
were made up of obscurely symmetrical curves and angles based on the
quantity of five. - H. P. Lovecraft
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Addenda: I got an email from Craig Kaplan, whose wonderful work on
tilings I mentioned above. He writes:
Because of the content of your post, I can't help but offer a few
notes about what you said. Feel free to use these any way you
want, or file them away for later.
* I wouldn't say that the Timurids set out to tackle fivefold tilings.
They looked at a lot of geometry in general - it's not clear to
me that they devoted any more energy to 5 than any other number.
But they did produce amazing results!
* You should be aware that within the Islamic geometric art
community, there's a fair amount of controversy and resentment
surrounding the Lu and Steinhardt paper. First, the paper
contains very strong claims that aren't supported by evidence.
Even if the artisans had some understanding of inflation (which
is debatable), I don't think there's any way they would have had
a notion of quasiperiodicity. Second, several researchers
perceive that L&S muscled their way into unfamiliar territory
without really finding out what had been done before - one could
argue that most of the work in their paper was well known to the
community. Finally, the paper made its mark not because of the
originality of its contribution, but because Science rolled out
an enormous publicity machine around the paper's release. This
is something that academics can't really control for, and which
I still find a bit baffling.
* Man, I'd also love to get my hands on Necipoglu's book on The
Topkapi Scroll. I knew of the book when it was in print, and
didn't buy it.
* Cromwell's article was in part a response to Lu & Steinhardt's.
You also might be interested in three upcoming articles of his,
to appear in the Journal of Mathematics and the Arts (for which
I'm an associated editor):
Islamic geometric designs from the Topkapi Scroll I:
Unusual arrangements of stars.
Islamic geometric designs from the Topkapi Scroll II:
A modular design system.
Hybrid 1-point and 2-point constructions for some Islamic
geometric designs.
Hopefully they'll be out soon.
In the meantime, I might also add that I did a bit of work on
understanding the origin of strange tilings like the one you
show with decagons, pentagons, and funky hexagons. It's in
this paper, which you didn't link to:
52) Craig S. Kaplan, Islamic star patterns from polygons in
contact, in GI '05: Proceedings of the 2005 conference on Graphics
Interface, 2005. Also available at
http://www.cgl.uwaterloo.ca/~csk/papers/gi2005.html
Hope that's useful to you, and thanks for the mention.
Brian Wichmann pointed out this online database:
53) Brian Wichmann, A tiling database, http://www.tilingsearch.org
Michael D. Hirschhorn emailed me to say that nearly 30 years ago, he
and David C. Hunt published a paper in the Journal of Combinatorial
Theory classifying all tilings of the plane by identical convex
equilateral pentagons. The most famous appears to be the "Hirschhorn
medallion, with angles 100-140-60-160-80.
Later Hirschhorn and Hunt extended their result to cover all
non-convex equilateral tilings, but this has never been published.
Presumably this page is based on Hirschhorn and Hunt's work:
54) MathPuzzle, The 14 different types of convex pentagons that
tile the plane, available at http://www.mathpuzzle.com/tilepent.html
For more discussion visit the n-Category Cafe at:
http://golem.ph.utexas.edu/category/2009/09/this_weeks_finds_in_mathematic_42.html
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