- #1
John Baez
[SOLVED] This Week's Finds in Mathematical Physics (Week 225)
Also available as http://math.ucr.edu/home/baez/week225.html
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... got to 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/
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
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
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
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 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.
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
Also available as http://math.ucr.edu/home/baez/week225.html
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... got to 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/
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
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
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
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 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.
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
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