# August 30, 2008 {#week269}

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
["Week 266"](#week266) and ["Week 268"](#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 the top view here is sulfur spewed
out by Pele, the biggest volcano on Io:

::: {align="center"}
[![](io_three_views_small.jpg)](io_three_views.jpg)
:::

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 spew sulfur dioxide up to 500 kilometers
high! Here's a picture of two such geysers, taken by the Galileo
spacecraft in 1997:

::: {align="center"}
[![](io_prometheus_plume.jpg){width="500"}](io_prometheus_plume.jpg)
:::

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

At the top horizon you can see a cloud of bluish-white haze. It may not
look big, but remember: Io is about 3600 kilometers in diameter, roughly
the size of our Moon. So, that hazy cloud is actually *huge*. It's a
geyser plume rising 140 kilometers above a massive volcano called Pillan
Patera. If you look carefully you can also see the Prometheus plume,
dead center, as a blue-grey ring. This plume 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.

::: {align="center"}
![](Galilean_moon_Laplace_resonance_animation.gif){width="300"}
:::

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](http://en.wikipedia.org/wiki/Allotropes_of_sulfur). 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":

::: {align="center"}
![](water_cube.jpg)
:::

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 design of this building is based on the Weaire-Phelan structure. But
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!

::: {align="center"}
![](honeycomb_cell_rhombic_dodecahedron.jpg)
:::

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 Tth had already proved it's true if we
assume the cells are polygons:

7) L. Fejes Tth, Regular Figures, Macmillan, 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!

::: {align="center"}
![](optimal_bubble_clusters.jpg)
:::

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

9) L. Fejes Tth, 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 a corner of a [rhombic
dodecahedron](http://en.wikipedia.org/wiki/Rhombic_dodecahedron):

::: {align="center"}
![](180px-Rhombicdodecahedron.gif)
:::

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](http://en.wikipedia.org/wiki/Truncated_octahedron):

::: {align="center"}
![](truncated_octahedron.gif)
:::

Using this, Tth found a design for the end of a honeycomb cell that
would be more efficient than what bees use:

::: {align="center"}
![](honeycomb_cell_truncated_octahedron.jpg)
:::

How much more efficient? How much area did Tth 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.

Tth 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:

::: {align="center"}
![](440px-Bitruncated_Cubic_Honeycomb.svg.png){width="400"}
:::

9) Wikipedia, Bitruncated cubic honeycomb,
`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:

10) 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:

::: {align="center"}
![](320px-12-14-hedral_honeycomb.png)
:::

11) Wikipedia, Weaire-Phelan structure,
[http://en.wikipedia.org/wiki/Weaire--Phelan_structure](http://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_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 the
ocean there's somewhere between 500 and 2500 billion tons of [methane
hydrate](http://en.wikipedia.org/wiki/Methane_clathrate), 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 it's an example of a "type I" gas hydrate,
where water molecules form cages patterned after the Weaire-Phelan
structure!

::: {align="center"}
![](gas_hydrate_structures.jpg)
:::

12) Wikipedia, Clathrate hydrate,
`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 structure 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:

13) 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:

14) Wikipedia, Convex uniform honeycomb,
`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
["Week 267"](#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....

------------------------------------------------------------------------

**Addenda:** I thank Jim Stasheff and Mike Stay for drawing my attention
to the Water Cube and its use of the Weaire-Phelan structure. I thank
Blake Stacey for catching a mistake.

------------------------------------------------------------------------

*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, [Aepinus
Atomized](http://zapatopi.net/kelvin/papers/aepinus_atomized.html)

*One word characterizes the most strenuous effors for the advancement of
science I have made perserveringly during fifty-five years; that word
is* Failure*. I know no more of electric or magnetic force, or of the
relation between ether, electricity and ponderable matter, or of
chemical affinity, than I knew and tried to teach to my students of
natural philosophy fifty years ago in my first session as professor.* -
Lord Kelvin, at a celebration of his life's work attended by more than
2000 guests