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December 24, 2005

This Week's Finds in Mathematical Physics - Week 225

John Baez

Happy holidays! I'll start with some gift suggestions for people

who put off their Christmas shopping a bit too late, 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?

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.)

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/ [Broken]

Crystal model of the Milky Way, http://www.bathsheba.com/crystal/galaxy/

My computer guru David Scharffenberg got me the 100-megaparsec cube,

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.

David says 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 [Broken]

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 about

200 years, 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 let 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 - a Wolfram Web Resource,

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 theta

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 - a Wolfram Web Resource,

http://mathworld.wolfram.com/Helicoid.html

In 1987 a fellow named Hoffman discovered a bunch more complete

non-self-intersecting minimal surface 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 [Broken]

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

calculation. 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.

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 [Broken]

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 [Broken]

I guess you 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 people have completed the classification, though.

Anyway, Schoen's "gyroid" is one of these triply periodic minimal

surfaces - apparently discovered before the computer revolution kicked in:

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 - a Wolfram Web Resource,

http://mathworld.wolfram.com/Gyroid.html

Apparently the gyroid is 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. 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.

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# This Week's Finds in Mathematical Physics (Week 225)

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