Also available at http://math.ucr.edu/home/baez/week269.html

August 30, 2008
This Week's Finds in Mathematical Physics (Week 269)
John Baez

No fancy math today.  I've been working hard with Aristide Baratin,
Laurent Freidel and Derek Wise on infinite-dimensional representations
of 2-groups.  It's a gnarly mix of higher category theory and analysis.
But I won't bother you with that until the paper is done.  I need a
break!  

So, today I want to talk about the sulfur geysers on Io, honeycombs,
the work of Kelvin, the Weaire-Phelan structure, and gas clathrates.

I already teased you with pictures of Jupiter's moon Io in "week266"
and "week268".  There's a reason.  Tortured by powerful tidal forces,
Io is the most geologically active object in the Solar System!  It
has mountains taller than Mount Everest, and over 400 active volcanos.
These put out the hottest lava ever seen - and a lot of it, too.  A
big eruption in 1997 produced more than 3500 square kilometers of the
stuff!

Most moons in the outer Solar System are pale and icy.  Io looks like
an evil pizza.  The ghastly red ring in one of the views here is
sulfur spewed out by Pele, the biggest volcano on Io:

1) NASA Photojournal, Three views of Io,
http://photojournal.jpl.nasa.gov/catalog/PIA00292

Io is big on sulfur.  It has lakes of molten sulfur... pale sulfur
dioxide snow... and geysers that spew plumes of sulfur dioxide up to
500 kilometers high!  Here's a picture of two such geysers, taken by
the Galileo spacecraft in 1979:

2) Astronomy Picture of the Day, Io: the Prometheus Plume,
http://apod.nasa.gov/apod/ap070211.html

At top you see a bluish plume rising 140 kilometers above a massive
volcano called Pillan Patera.  If you look carefully you can also see
the Prometheus plume, dead center.  This thing has been active at least
since 1979 - but it's *moved 85 kilometers west* during that time!  
Scary.

Ironically, all this chaotic activity on Io may be caused by the
"music of the spheres".  Io is locked in a 2:1 orbital resonance
with the moon Europa, and a 4:1 resonance with Ganymede.  These keep
Io's orbit a bit eccentric, which causes tidal heating - to the tune
of about 100 trillion watts.

Another interesting thing is that the red spots on Io are made of
sulfur, but so are the yellow plains.  Of course there are lots of
compounds involving sulfur, but even the pure element has a lot of
different forms, or "allotropes".  I've always been fascinated by
those.

Back here on Earth, the Weaire-Phelan structure was in the news
recently!  If you watched the Olympics in Beijing, you may have
seen the National Aquatics Center - a building also called the
"Water Cube":

3) Spectacular mathematical bubble design at the Olympics, Math in
the News, Mathematical Association of America, August 8, 2008,
http://mathdl.maa.org/mathDL?pa=mathNews&sa=view&newsId=392

The story behind the design of this building goes back to 36 BC,
when Marcus Terentius Varro described two competing theories for
why bees have hexagonal honeycombs.  The first said: because bees
have six legs!  The second said: efficiency!  You see, a hexagonal
lattice lets you divide the plane into cells of equal area with
the least possible perimeter per cell.  So if the bees want to save
wax, that's the pattern they'll pick.

The second theory seems more plausible.  But is it true?  I'm not even
sure how to resolve that question.  But it's worth noting that
honeycomb cells are actually 3-dimensional - and the *end* of each
cell consists of three rhombi that meet at the same angle as bubbles
in soap suds!  

Now, soap films minimize surface area subject to whatever constraints
they encounter.  So, a single bubble that holds a given amount of
air will form a sphere.  But soap suds with lots of bubbles do more
complicated things.  Take a bubble bath and pay careful attention!  
You'll see that three bubble faces meet along each edge, at precisely
120 degree angles.  And when four bubbles meet at a vertex, they form
a pattern with tetrahedral symmetry, with edges meeting at an angle
of arccos(-1/3), or about 109.5 degrees.

So if honeycombs display these patterns, we can guess area is being
minimized.

But as usual, things get more complicated when you look deeper.  First,
while everyone *believed* for a long time that a hexagonal honeycomb
is the way to divide the plane into equal-area cells with minimal
perimeters, this was only *proved* much later: in 1999, by Thomas
Hales.

It's an interesting story.  Hales had just finished his epic proof of
Kepler's conjecture about the densest way to pack equal-sized spheres -
a proof so complicated that the referees "ran out of energy" trying
to check it.  That's quite a tale in itself... but to avoid an
infinite sequence of nested digressions, I'll refer you to these:

4) George G. Szpiro, Kepler's Conjecture, John Wiley and Sons, 2003.  
Reviewed by Frank Morgan in Notices Amer. Math. Soc. 52 (2005), 44-47.
Also available at http://www.ams.org/notices/200501/rev-morgan.pdf

5) Thomas Hales, The Kepler Conjecture,
http://www.math.pitt.edu/~thales/kepler98/

Anyway, after Hales proved the Kepler Conjecture, Denis Weaire
suggested that he tackle the Hexagonal Honeycomb Conjecture - and
Hales promptly solved that too!  He said, "In contrast with the years
of forced labor that gave the Kepler Conjecture, I felt as if I had
won the lottery."

6) Thomas C. Hales, The Honeycomb Conjecture,
http://www.math.pitt.edu/~thales/kepler98/honey/

Hales wasn't the first to make progress on the Hexagonal Honeycomb
Conjecture.  A guy named Fejes Tóth had already proved it's true if we
assume the cells are polygons:

7) L. Fejes Tóth, Regular Figures, Macmillan Co, New York, 1964.

So what Hales had to do is rule out cells with curved edges.  This
is harder than you might think.  In fact, for clusters of finitely
many cells, the optimal shapes can be curved, even near the middle!
You can see pictures in the review by Frank Morgan above, or here:

8) S. J. Cox, M. Fatima Vas, C. Monnereua-Pittet and N. Pittet,
Minimal perimeter for N identical bubbles in two dimensions:
calculations and simulations, Phil. Mag. 83 (2003), 1393-1406.

Another thing Tóth did is carefully define the 3d optimization
problem that bees might be trying to solve, and find a slightly
better solution:

9) L. Fejes Tóth, What the bees know and what the bees do not know,
Bull. Amer. Math. Soc. 70 (1964), 468-481.   Also available at
http://projecteuclid.org/euclid.bams/1183526078

I won't try to describe his results in detail, since the paper is
freely available and well-written.  But here's the basic idea.  The
end of a bee's honeycomb cell looks just like the corner of a rhombic
dodecahedron.  This is a 12-sided solid that you can pack to
completely fill space.  This makes sense, because the ends of
one layer of cells in a honeycomb should neatly fit against those
of the next layer.  

However, it's been known since the work of Kelvin that there's another
solid you can use to pack space more efficiently: that is, with less
surface area per cell.  This is the truncated octahedron.  Using
this, Tóth found a design for the end of a honeycomb cell that would
be more efficient than what bees use!  

How much more efficient?  How much area did Tóth manage to shave off?
Almost 0.35% of the area of cell's opening!  In the eternal battle
of man against bee, we triumph yet again!  It makes me proud to be
human.

Tóth is more modest:

   We must admit that all this has no practical consequence....  
   Besides, the building style of the bees is definitely simpler
   than that described above.  So we would fail in shaking someone's
   conviction that the bees have a deep geometrical intuition.

I doubt "intuition" is the right word for it, but they're definitely
good at what they do.

Now, back to Kelvin!  When he bumped into the truncated octahedron,
he was actually studying the 3d version of the 2d Hexagonal Honeycomb
Conjecture.  In other words, he was trying to chop 3d space into cells
of equal volume with the least surface area per cell.  And he
conjectured that the answer was very similar to filling space with
truncated octahedra, as shown here:

10) Bitruncated cubic honeycomb, Wikipedia,
http://en.wikipedia.org/wiki/Bitruncated_cubic_honeycomb

I say "very similar" because it's actually more efficient if you let
the hexagonal faces in this structure be slightly *curved*.  So, the
possibility Hales ruled out in the 2d case actually matters here!
In his 1887 paper on this subject, Kelvin wrote:

   No shading could show satisfactorily the delicate curvature of
   the hexagonal faces, though it may be fairly well seen on the
   solid model made as described in Section 12.  But it is shown  
   beautifully, and illustrated in great perfection, by making a
   skeleton model of 36 wire arcs for the 36 edges of the complete
   figure, and dipping it in soap solution to fill the faces with
   film, which is easily done for all the faces but one.  The
   curvature of the hexagonal film on the two sides of the plane
   of its six long diagonals is beautifully shown by reflected light.

I think this is a nice passage.  We may remember Kelvin for his
profound work on electromagnetism and thermodynamics - or his 1900
lecture on two "dark clouds" hanging over physics: the Michelson-
Morley experiment (which foreshadowed special relativity) and black
body radiation (which foreshadowed quantum mechanics).  We may not
imagine him playing around with soap bubbles!  But it shows that good
science stems from curiosity, and curiosity knows no bounds.

You can read Kelvin's paper here:

11) Lord Kelvin, On the division of space with minimum partitional
area, Phil. Mag. 24 (1887), 503.  Also available at
http://zapatopi.net/kelvin/papers/on_the_division_of_space.html

His lively mind is evident from the selection of papers on this site.
For example: "On Vortex Atoms", where he unsuccessfully tried to build
atoms out of knotted electromagnetic field lines, and wound up giving
birth to knot theory.  Some others I hadn't heard of: "On the origin
of life", "The sorting demon of Maxwell", and "Windmills must be the
future source of power".

Anyway: for over a century the so-called "Kelvin structure" was
believed to be the best solution to the problem of chopping space
into equal volume cells with minimal surface area.   But in 1993
two physicists at Trinity College in Dublin - Denis Weaire and
Robert Phelan - found a solution that has 0.3% less surface area!  
It looks like this:

12) Weaire-Phelan structure, Wikipedia,
http://en.wikipedia.org/wiki/Weaire-Phelan_structure

It's built from *two* kinds of cells - a 12-sided one and a 14-sided
one.  Was that allowed in Kelvin's original puzzle?  I can't tell!

In fact, this so-called "Weaire-Phelan structure" was no bolt out
of the blue.  The basic pattern had already been seen in certain
cage-like crystals called "clathrates".  For example, down at the
bottom of cold oceans, there are about 6 trillion tons of "methane
hydrate", a funny substance in which methane molecules are trapped
in polyhedral cages formed by water molecules.  It looks like ice,
but you can ignite it with a cigarette lighter!  If you think global
warming is bad now, just wait until people figure out how to mine
this stuff....  

Anyway, methane hydrate is just one of a collection of gas hydrates
with different geometries.  And in a so-called "type I" gas hydrate,
the water molecules form cages in the pattern of the Weaire-Phelan
structure!

13) Clathrate hydrate, Wikipedia,
http://en.wikipedia.org/wiki/Clathrate_hydrate

So, it's nicely appropriate to use the Weaire-Phelan structure for a
building called the Water Cube!  Since this struture is perfectly
periodic, the engineer for the Water Cube cut it at an odd angle to
make it look more exciting.  The MAA article cited above says:

   Made of a plastic known as ethylene tetrafluoroethylene and
   filled with air, the bubbles are attached to a steel framework
   outlining the bubble edges.  Surface tension holds the bubbles
   together and tends to pull them into a structure with least
   surface area.

   The building "really looks like nothing else in the world,"
   Tristam Carfrae told the New York Times. "It's a box made of
   bubbles."  Carfrae is the structural engineer who designed the
   center.

On the math side of things, there's plenty left to be done.  Nobody
has proved that the Weaire-Phelan structure is the best solution
to Kelvin's problem.  According to Frank Morgan, the expert
on minimal surface who reviewed Spziro's book on Kepler's problem,

   Proving the Weaire-Phelan structure optimal looks perhaps a
   century beyond current mathematics to me, but I understand that
   Hales is already thinking about it.

More generally, minimal surface theory is a lively subject that
uses a lot of deep tools.  Morgan is really big on explaining math,
so his book is probably the place to start if you want to dig deeper:

14) Frank Morgan, Geometric Measure Theory: a Beginner's Guide,
Academic Press, New York, 2000.

Personally, I'm more in love with symmetry than minimization.
So, I want to learn more about the 28 "convex uniform honeycombs" -
ways of uniformly packing 3d space with uniform solids:

15) Convex uniform honeycomb, Wikipedia,
http://en.wikipedia.org/wiki/Convex_uniform_honeycomb

They're related to Coxeter groups and "crystallographic groups",
which are the 3d analogues of the wallpaper groups I discussed
back in "week267".  In fact, we can study honeycombs and their
symmetry groups in any dimension, both in flat space and in
positively curved (spherical) and negatively curved (hyperbolic)
space.  A lot is known about them....

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Quote of the Week:

   My suggestion is that Aepinus' fluid consists of exceedingly minute
   equal and similar atoms, which I call electrions, much smaller than
   the atoms of ponderable matter.

          Lord Kelvin, from Aepinus Atomized

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