Happy holidays! I'll start with some gift suggestions for people who put off their Christmas shopping a bit too long, before moving on to this week's astronomy pictures and then some mathematical physics: minimal surfaces.
Back in 2000 I listed some gift ideas in "week162". I decided to do it again this year. After all, where else can you read about quantum gravity, nonabelian cohomology, higher categories... and also get helpful shopping tips? But, I put off writing this Week's Finds a bit too late. Oh well.
I just saw this book in a local store, and it's great:
1) Robert Dinwiddie, Philip Eales, David Hughes, Ian Nicholson, Ian Ridpath, Giles Sparrow, Pam Spence, Carole Stott, Kevin Tildsley, and Martin Rees, Universe, DK Publishing, New York, 2005.
If you like the astronomy pictures you've seen here lately, you'll love this book, because it's full of them - all as part of a well-organized, clearly written, information-packed but nontechnical introduction to astronomy. It starts with the Solar System and sails out through the Oort Cloud to the Milky Way to the Local Group to the Virgo Supercluster ... and all the way out and back to the Big Bang!
The only thing this book seems to lack - though I could have missed it - is a 3d map showing the relative scales of our Solar System, Galaxy, and so on. I recommended a wall chart like this back in "week162", and my friend Danny Stevenson just bought me one. I'll probably put it up near my office in the math department... gotta keep the kids thinking big!
You don't really need to buy a chart like this. You can just look at this website:
2) Richard Powell, An Atlas of the Universe, http://www.anzwers.org/free/universe/
It has nine maps, starting with the stars within 12.5 light years and zooming out repeatedly by factors of 10 until it reaches the limits of the observable universe, roughly 14 billion light years away. Or more precisely, 14 billion years ago!
(The farther we look, the older things we see, since light takes time to travel. The most distant thing we see is light released when hot gas from the Big Bang cooled down just enough to let light through! If we calculate how far this gas would be now, thanks to the expansion of the universe, we get a figure of roughly 78 billion light years. But of course we can't see what that gas looks like now unless we wait a lot longer. It's a bit confusing until you think about it for a while.)
For example, here are the clusters of galaxies within 100 million lightyears of us:
The biggest of these is the Virgo cluster, which I discussed in "week224". This contains about 2000 galaxies. The second biggest is the Fornax cluster. The whole agglomeration shown here is called the Virgo Supercluster. Superclusters are among the biggest structures in the universe.
This atlas is fun to browse when you're at your computer. But, if someone you know wants to contemplate the universe in a more relaxing way, try getting them one of these:
3) Bathsheba Grossman, Crystal model of a typical 100-megaparsec cube of the universe, http://www.bathsheba.com/crystal/largescale/
Crystal model of the Milky Way, http://www.bathsheba.com/crystal/galaxy/
I found out about these from David Scharffenberg, who owns the Riverside Computer Center nearby - a cool little shop that's decorated with archaic technology ranging from a mammoth slide rule to a gizmo that computes square roots using air pressure. He gave me the 100-megaparsec cube as a present, and it's great! It's lit up from below, and it shows the filaments, sheets and superclusters of galaxies that reign supreme at this distance scale. 100 megaparsecs is about 300 million light years, so this view is a bit bigger than the previous picture:
David says Grossman's model of the Milky Way is also nice: it takes into account the latest research, which shows our galaxy is a "barred" spiral! You can see the bar in the middle here:
4) R. Hurt, NASA/JPL-Caltech, Milky Way Bar, http://www.spitzer.caltech.edu/Media/mediaimages/sig/sig05-010.shtml
If you really have money to burn, Grossman has also made nice sculptures of mathematical objects like the 24-cell, the 600-cell and Schoen's gyroid - a triply periodic minimal surface that chops 3-space into two parts:
5) Bathsheba Grossman, Mathematical models, http://www.bathsheba.com/math/
However, the great thing about the web is that lots of beautiful stuff is free - like these pictures of the gyroid:
I explained the 24-cell and 600-cell in "week155". So, let me explain the gyroid - then I need to start cooking up a Christmas eve dinner!
A "minimal surface" is a surface in ordinary 3d space that can't reduce its area by changing shape slightly. You can create a minimal surface by building a wire frame and then creating a soap film on it. As long as the soap film doesn't actually enclose any air, it will try to minimize its area - so it will end up being a minimal surface.
If you make a minimal surface this way, it will have edges along the wire frame. A minimal surface without edges is called "complete". For a long time, the only known complete minimal surfaces that didn't intersect themselves were the plane, the catenoid, and the helicoid. You get a "catenoid" by taking an infinitely long chain and letting it hang to form a curve called a "catenary"; then you use this curve to form a surface of revolution, which is the catenoid:
6) Eric Weisstein, Catenoid, from Mathworld, http://mathworld.wolfram.com/Catenoid.html
In cylindrical coordinates the catenoid is given by the equation
r = c cosh(z/c)
for your favorite constant c.
A "helicoid" is like a spiral staircase; in cylindrical coordinates it's given by the equation
z = c θ
for some constant c. You can see a helicoid here - and see how it can continuously deform into a catenoid:
7) Eric Weisstein, Helicoid, from Mathworld, http://mathworld.wolfram.com/Helicoid.html
In 1987 a fellow named Hoffman discovered a bunch more complete non-self-intersecting minimal surfaces with the help of a computer:
8) D. Hoffman, The computer-aided discovery of new embedded minimal surfaces, Mathematical Intelligencer 9 (1987), 8-21.
Since then people have gotten good at inventing minimal surfaces. You can see a bunch here:
9) GRAPE (Graphics Programming Environment), Surface overview, http://www-sfb256.iam.uni-bonn.de/grape/EXAMPLES/AMANDUS/bmandus.html
10) GANG (Geometry Analysis Numerics Graphics), Gallery of minimal surfaces, http://www.gang.umass.edu/gallery/min/
As you can see, people who work on mininal surfaces like goofy acronyms. If you look at the pictures, you can also see that a minimal surface needs to be locally saddle-shaped. More precisely, it has "zero mean curvature": at any point, if it curves one way along one principal axis of curvature, it has to curve an equal and opposite amount along the perpendicular axis. Supposedly this was proved by Euler.
If we write this requirement as an equation, we get a second-order nonlinear differential equation called "Lagrange's equation" - a special case of the Euler-Lagrange equation we get from any problem in the variational calculus. So, finding new minimal surfaces amounts to finding new solutions of this equation. Soap films solve this equation automatically, but only with the help of a wire frame; it's a lot more work to find minimal surfaces that are complete.
For the theoretical physicist, minimal surfaces also go by another name: strings! The "worldsheet" of a bosonic string is just a 2-dimensional surface in spacetime. The equation governing the string's motion just says that the area of this surface can't be reduced by wiggling it slightly. In other words, it's just Lagrange's equation. There's a big difference between string theory and the theory of minimal surfaces, though: in string theory we need to take quantum mechanics into account! (Another big difference is that spacetime is a Lorentzian rather than Riemannian manifold, unless we do a trick called "Wick rotation".)
So, bosonic string theory is about the quantum version of soap films - and "D-branes" serve as the wire frames. But if this reminds you of "spin foams", I should warn you: there are a few big differences. The main thing is that spin foams are background-free: they don't live in spacetime, they are spacetime. So, it doesn't make any obvious sense for them to minimize area, though Smolin has suggested it might make an unobvious kind of sense. All the fun must happen when the "bubbles" of a spin foam meet along their edges... but we don't really know how this should work, to create a foam with the right consistency at large scales.
There are a lot of minimal surfaces that have periodic symmetry in 3 directions, like a crystal lattice. You can learn about them here:
11) Elke Koch, 3-periodic minimal surfaces, http://staff-www.uni-marburg.de/~kochelke/minsurfs.htm
In fact, they have interesting relations to crystallography:
12) Elke Koch and Werner Fischer, Mathematical crystallography http://www.staff.uni-marburg.de/~kochelke/mathcryst.htm#minsurf
I guess people can figure out which of the 230 crystal symmetry groups (or "space groups") can arise as symmetries of triply periodic minimal surfaces, and use this to help classify these rascals. A cool mixture of group theory and differential geometry! I don't get the impression that they have completed the classification, though.
Anyway, Schoen's "gyroid" is one of these triply periodic minimal surfaces. Schoen discovered it before the computer revolution kicked in. He was working for NASA, and his idea was to use it for building ultra-light, super-strong structures:
13) A. H. Schoen, Infinite periodic minimal surfaces without selfintersections, NASA Tech. Note No. D-5541, Washington, DC, 1970.
You can learn more about the gyroid here:
14) Eric Weisstein, Gyroid, from Mathworld, http://mathworld.wolfram.com/Gyroid.html
Apparently it's the only triply periodic non-self-intersecting minimal surface with "triple junctions". I'm not quite sure what that means mathematically, but I can see them in the picture!
I said that soap films weren't good at creating complete minimal surfaces. But actually, people have seen at least approximate gyroids in nature, made from soap-like films:
15) P. Garstecki and R. Holyst, Scattering patterns of self-assembled gyroid cubic phases in amphiphilic systems, J. Chem. Phys. 115 (2001), 1095-1099.
An "amphiphilic" molecule is one that's attracted by water at one end and repelled by water at the other. For example, the stuff in soap. Mixed with water and oil, such molecules form membranes, and really complicated patterns can emerge, verging on the biological. Sometimes the membranes make a gyroid pattern, with oil on one side and water on other! It's a great example of how any sufficiently beautiful mathematical pattern tends to show up in nature somewhere... as Plato hinted in his theory of "forms".
People have fun simulating these "ternary amphiphilic fluids" on computers:
16) Nelido Gonzalez-Segredo and Peter V. Coveney, Coarsening dynamics of ternary amphiphilic fluids and the self-assembly of the gyroid and sponge mesophases: lattice-Boltzmann simulations, available as cond-mat/0311002.
17) Pittsburgh Supercomputing Center, Ketchup on the grid with joysticks, http://www.psc.edu/science/2004/teragyroid/
The second site above describes the "TeraGyroid Project", in which people used 17 teraflops of computing power at 6 different facilities to simulate the gyroidal phase of oil/water/amphiphile mixtures and study how "defects" move around in what's otherwise a regular pattern. The reference to ketchup comes from some supposed relationship between these ternary amphiphilic fluids and how ketchup gets stuck in the bottle. I'm not sure ketchup actually is a ternary amphiphilic fluid, though!
Hmm. I just noticed a pattern to the websites I've been referring to: first one about a "Milky Way bar", then one about a "GRAPE", and now one about ketchup! I think it's time to cook that dinner.
© 2005 John Baez