# November 20, 2006 {#week241}

I've been working too hard, and running around too much, to write This
Week's Finds for a while. A bunch of stuff has built up that I want to
explain. Luckily I've been running around explaining stuff --- higher
gauge theory, and tales of the dodecahedron.

This weekend I went to Baton Rouge. I was invited to Louisiana State
University by Jorge Pullin of loop quantum gravity fame, and I used the
opportunity to get a look at LIGO --- the Laser Interferometry
Gravitational-Wave Observatory!

I described this amazing experiment back in ["Week 189"](#week189),
so I won't rehash all that. Suffice it to say that there are two
installations: one in Hanford Washington, and one in Livingston
Louisiana. Each consists of two evacuated tubes 4 kilometers long,
arranged in an L shape.

Laser beams bounce back and forth between mirrors suspended at the ends
of the tubes, looking for tiny changes in their distance that would
indicate a gravitational wave passing through, stretching or squashing
space. And when I say "tiny", I mean smaller than the radius of a
proton! This is serious stuff.

Jorge drove me in his SUV to Livingston, a tiny town about 20 minutes
from Baton Rouge. While he runs the gravity program at Louisiana State
University, which has links to LIGO, he isn't officially part the LIGO
team. His wife is. When I first met Gabriela Gonzalez, she was studying
the Brownian motion of torsion pendulums. The mirrors in LIGO are hung
on pendulums made of quartz wire, to minimize the effect of vibrations.
But, the random jittering of atoms due to thermal noise still affects
these pendulums. She was studying this noise to see its effect on the
accuracy of the experiment.

This was way back when LIGO was just being planned. Now that LIGO is a
reality, she's doing data analysis, helping search for gravitational
waves produced by pairs of neutron stars and/or black holes as they
spiral down towards a sudden merger. Together with an enormous pageful
of authors, she helped write this paper, based on data taken from the
"first science run" --- the first real LIGO experiment, back in 2002:

1) The LIGO Scientific Collaboration, "Analysis of LIGO data for gravitational waves from binary neutron stars", _Phys. Rev._ **D69** (2004), 122001. Also available at [`gr-qc/0308069`](https://arxiv.org/abs/gr-qc/0308069).

She's one of the folks with an intimate knowledge of the experimental
setup, who keeps the theorists' feet on the ground while they stare up
into the sky.

On the drive to Livingston, Jorge pointed out the forests that surround
the town. These forests are being logged:
$$\includegraphics[max width=0.65\linewidth]{../images/ligo_0.jpg}$$
I asked him about this --- when I last checked, the vibrations from
falling trees were making it impossible to look for gravitational waves
except at night! He said they've added a "hydraulic external
pre-isolator" to shield the detector from these vibrations --- basically
a super-duper shock absorber:
$$\includegraphics[max width=0.65\linewidth]{../images/hepi.jpg}$$
Now they can operate LIGO day and night.

I also asked him how close LIGO had come to the sensitivity levels they
were seeking. When I wrote ["Week 189"](#week189), during the first
science run, they still had a long way to go. That's why the above
paper only sets upper limits on neutron star collisions within 180
kiloparsecs. This only reaches out to the corona of the Milky Way ---
which includes the Small and Large Magellanic Clouds. We don't expect
many neutron star collisions in this vicinity: maybe one every 3 years
or so. The first science run didn't see any, and the set an upper limit
of about 170 per year: the best experimental upper limit so far, but
definitely worth improving, and nowhere near as fun as actually *seeing*
gravitational waves.

But Jorge said the LIGO team has now reached its goals: they should be
able to see collisions out to 15 megaparsecs! By comparison, the center
of the Virgo cluster is about 20 megaparsecs away. In fact, they should
already be able to see about half the galaxies in this cluster.

They're now on their seventh science run, and they'll keep upping the
sensitivity in future projects called "Enhanced LIGO" and "Advanced
LIGO". The latter should see neutron star collisions out to 300
megaparsecs:

2) Advanced LIGO, `http://www.ligo.caltech.edu/advLIGO/`

When we arrived at the gate, Jorge spoke into the intercom and got us
let in. Our guide, Joe Giaime, was running a bit a late, so we walked
over and looked at the interferometer's arms, each of which stretched
off beyond sight, 2.5 kilometers of concrete tunnel surrounding the
evacuated piping --- the world's largest vacuum facility:
$$\includegraphics[max width=0.65\linewidth]{../images/ligo_1.jpg}$$
Schoolkids have been invited to paint pictures on some of these pipes:
$$\includegraphics[max width=0.65\linewidth]{../images/ligo_2.jpg}$$

One can tell this is the South. The massive construction caused pools of
water to form in the boggy land near the facility, and these pools then
attracted alligators. These have been dealt with firmly. The game
hunters who occasionally fired potshots at the facility were treated
more forgivingly: instead of feeding them to the alligators, the LIGO
folks threw a big party and invited everyone from the local hunting
club. Hospitality works wonders down here.

The place was pretty lonely. During the week lots of scientists work
here, but this was Saturday, and on weekends there's just a skeleton
crew of two. There's usually not much to do now that the experiment is
up and running. As Joe Giaime later said, there have been no "Jodie
Foster moments" like in the movie Contact, where the scientists on duty
suddenly see a signal, turn on the suspenseful background music, and
phone the President. There's just too much data analysis required to
see any signal in real time: data from both Livingston and Hanford is
sent to Caltech, and then people grind away at it. So, about the most
exciting thing that happens is when the occasional tiny earthquake
throws the laser beam out of phase lock.

When Joe showed up, I got to see the main control room, which is dimly
lit, full of screens indicating noise and sensitivity levels of all
sorts --- and even some video monitors showing the view down the laser
tube:
$$\includegraphics[max width=0.65\linewidth]{../images/ligo_3.jpg}$$
This is where the people on duty hang out --- you can see those video
monitors on top:
$$\includegraphics[max width=0.65\linewidth]{../images/ligo_4.jpg}$$
One of them had brought his sons, in a feeble attempt to dispose of the
huge supply of Halloween candy that had somehow collected here.

I also got to see a sample of the 400 "optical baffles" which have
been installed to absorb light spreading out from the main beam before
it can bounce back in and screw things up. The interesting thing is that
these baffles and their placement were personally designed by Kip Thorne
and some other godlike LIGO figure. Moral: unless they've gone soft,
even bigshot physicists like to actually think about physics now and
then, not just manage enormous teams.

But overall, there was surprisingly little to see, since the innermost
workings are all sealed off, in vacuum. The optics are far more
complicated than my description --- "a laser bouncing between two
suspended mirrors" --- could possibly suggest. But, all I got to see was
a chart showing how they work:
$$\includegraphics[max width=0.65\linewidth]{../images/ligo_5.jpg}$$
Oh well. I'm glad I don't need to understand this stuff in detail. It
was fun to get a peek.

By the way, I wasn't invited to Louisiana just to tour LIGO and eat
[beignets](http://www.crescentcitybeignets.com/beignet.html) and
[alligator sushi](http://www.servingsushi.com/CatSubCat/CatSubCatDisplay2.asp?p1=9&p2=Y&p7=2&p8=528&p9=CSC1).
My real reason for going there was to talk about higher gauge theory --- a
generalization of gauge theory which studies the parallel transport not
just of point particles, but also strings and higher-dimensional
objects:

3) John Baez, "Higher gauge theory", `http://math.ucr.edu/home/baez/highergauge`

This is a gentler introduction to higher gauge theory than my previous
talks, some of which I inflicted on you in ["Week 235"](#week235).
It explains how $BF$ theory can be seen as a higher gauge theory, and
briefly touches on Urs Schreiber's work towards exhibiting Chern-Simons
theory and $11$-dimensional supergravity as higher gauge theories. The
webpage has links to more details.

I was also travelling last weekend --- I went to Dartmouth and gave this
talk:

4) John Baez, "Tales of the Dodecahedron: from Pythagoras through Plato to Poincare", `http://math.ucr.edu/home/baez/dodecahedron/`

It's full of pictures and animations --- fun for the whole family!

I started with the Pythagorean fascination with the pentagram, and how
you can use the pentagram to give a magical picture proof of the
irrationality of the golden ratio.

I then mentioned how Plato used four of the so-called Platonic solids to
serve as atoms of the four elements --- earth, air, water and fire -
leaving the inconvenient fifth solid, the dodecahedron, to play the role
of the heavenly sphere. This is what computer scientists call a
"kludge" --- an awkard solution to a pressing problem. Yes, there are
twelve constellations in the Zodiac --- but unfortunately, they're
arranged quite differently than the faces of the dodecahedron.

This somehow led to the notion of the dodecahedron as an atom of
"aether" or "quintessence" --- a fifth element constituting the
heavenly bodies. If you've ever seen the science fiction movie The
Fifth Element, now you know where the title came from! But once upon a
time, this idea was quite respectable. It shows up as late as Kepler's
Mysterium Cosmographicum, written in 1596.

I then went on to discuss the $120$-cell, which gives a way of chopping a
spherical universe into 120 dodecahedra.
$$\includegraphics[max width=0.65\linewidth]{../images/Stereographic_polytope_120cell_faces.png}$$

This leads naturally to the Poincare homology sphere, a closely related
3-dimensional manifold made by gluing together opposite sides of *one*
dodecahedron.

The Poincare homology sphere was briefly advocated as a model of the
universe that could explain the mysterious weakness of the
longest-wavelength ripples in the cosmic background radiation --- the
ripples that only wiggle a few times as we scan all around the sky:

5) J.-P. Luminet, J. Weeks, A. Riazuelo, R. Lehoucq, and J.-P. Uzan, "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background", _Nature_ **425** (2003), 593. Also available as [`astro-ph/0310253`](https://arxiv.org/abs/astro-ph/0310253).

The idea is that if we lived in a Poincare homology sphere, we'd see
several images of each very distant point in the universe. So, any
ripple in the background radiation would repeat some minimum number of
times: the lowest-frequency ripples would be suppressed.

Alas, this charming idea turns out not to fit other data. We just don't
see the same distant galaxies in several different directions:

6) Neil J. Cornish, David N. Spergel, Glenn D. Starkman and Eiichiro Komatsu, "Constraining the topology of the universe", _Phys. Rev. Lett._ **92** (2004) 201302. Also available as [`astro-ph/0310233`](https://arxiv.org/abs/astro-ph/0310233).

For a good review of this stuff, see:

7) Jeffrey Weeks, "The Poincare dodecahedral space and the mystery of the missing fluctuations", _Notices of the AMS_ **51** (2004), 610--619. Also available at `http://www.ams.org/notices/200406/fea-weeks.pdf`

In the abstract of my talk, I made the mistake of saying that the
regular dodecahedron doesn't appear in nature --- that instead, it was
invented by the Pythagoreans. You should never say things like this
unless you want to get corrected!

Dan Piponi pointed out this dodecahedral virus:
$$\href{http://www.nature.com/nsmb/journal/v8/n1/pdf/nsb0101_77.pdf}{\includegraphics[max width=0.65\linewidth]{../images/pariacoto_virus.jpg}}$$

8) Liang Tang et al, "The structure of Pariacoto virus reveals a dodecahedral cage of duplex RNA", _Nature Structural Biology_ **8** (2001), 77--83. Also available at `http://www.nature.com/nsmb/journal/v8/n1/pdf/nsb0101_77.pdf`

The first black line is 100 angstroms long ($10^{-8}$ meters), while the
second is 50 angstroms long.

Garett Leskowitz pointed out the molecule "dodecahedrane", with 20
carbons at the vertices of a dodecahedron and 20 hydrogens bonded to
these:
$$\href{http://en.wikipedia.org/wiki/Image:Dodecahedran.png}{\includegraphics[max width=0.65\linewidth]{../images/Dodecahedran.png}}$$

9) Wikipedia, "Dodecahedrane", `http://en.wikipedia.org/wiki/Dodecahedrane`

This molecule hasn't been found in nature yet, but chemists can
synthesize it using reactions like these:
$$\href{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=346698}{\includegraphics[max width=0.65\linewidth]{../images/dodecahedrane_synthesis_1.jpg}}$$
$$\href{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=346698}{\includegraphics[max width=0.65\linewidth]{../images/dodecahedrane_synthesis_2.jpg}}$$

10) Robert J. Ternansky, Douglas W. Balogh and Leo A. Paquette, "Dodecahedrane", _J. Am. Chem. Soc._ **104** (1982), 4503--4504.

11) Leo A. Paquette, "Dodecahedrane --- the chemical transliteration of Plato's universe (a review)", _Proc. Nat. Acad. Sci. USA_ **14** part 2 (1982), 4495--4500. Also available at `http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=346698`

So, there's probably a bit somewhere in our galaxy.

Of course, what I *meant* was that people didn't come up with regular
dodecahedra after seeing them in nature --- that instead, the Pythagoreans
dreamt them up, possibly after seeing pyrite crystals that look sort of
similar. These crystals are called "pyritohedra".

But, even here I made a mistake. The Pythagoreans seem not to have been
the first to discover the dodecahedron. John McKay told me that stone
spheres with Platonic solids carved on them have been found in Scotland,
dating back to around 2000 BC!
$$\href{http://arxiv.org/abs/math-ph/0303071}{\includegraphics[max width=0.65\linewidth]{../images/blocks.png}}$$

13) Michael Atiyah and Paul Sutcliffe, "Polyhedra in physics, chemistry and geometry", available as [`math-ph/0303071`](http://arxiv.org/abs/math-ph/0303071).

14) Dorothy N. Marshall, "Carved stone balls", _Proc. Soc. Antiq. Scotland_ **108** (1976/77), 40--72. Available at `http://ads.ahds.ac.uk/catalogue/library/psas/`

Indeed, stone balls with geometric patterns on them have been found
throughout Scotland, and occasionally Ireland and northern England. They
date from the Late Neolithic to the Early Bronze age: 2500 BC to 1500
BC. For comparison, the megaliths at Stonehenge go back to 2500--2100 BC.

Nobody knows what these stone balls were used for, though the article by
Marshall presents a number of interesting speculations.

The pyritohedron is interesting in itself, so before I turn to some
really fancy math, let me talk a bit about this guy. Since pyrite is
fundamentally a cubic crystal, the pyritohedron is basically made out of
little cubic cells, as shown here:
$$\href{http://www.uwgb.edu/dutchs/symmetry/isometuc.htm}{\includegraphics[max width=0.65\linewidth]{../images/pyritohedron.png}}$$

12) Steven Dutch, "Building isometric crystals with unit cells", `http://www.uwgb.edu/dutchs/symmetry/isometuc.htm`

It has 12 pentagonal faces, orthogonal to these vectors:
$$
  \begin{gathered}
    (2,1,0) \quad (2,-1,0) \quad (-2,1,0) \quad (-2,-1,0)
  \\(1,0,2) \quad (-1,0,2) \quad (1,0,-2) \quad (-1,0,-2)
  \\(0,2,1) \quad (0,2,-1) \quad (0,-2,1) \quad (0,-2,-1)  
  \end{gathered}
$$

You can see how this works by going here:

13) mindat.org, "Pyrite", `http://www.mindat.org/min-3314.html`

If your webbrowswer can handle Java, go to this webpage and click on
"Pyrite no. 3" to see a rotating pyritohedron. Then, while holding
your left mouse button down when the cursor is over the picture of the
pyritohedron, type "m" to see the vectors listed above.

Why "m"? These vectors are called "Miller indices". In general,
Miller indices are outwards-pointing vectors orthogonal to the faces of
a crystal; we can use them to classify crystals.

The Miller indices for the pyritohedron have a nice property. If you
think of these 12 vectors as points in space, they're the corners of
three $2\times1$ rectangles: a rectangle in the $xy$ plane, a rectangle in the $xz$
plane, and a rectangle in the $yz$ plane.

These points are also corners of an icosahedron! It's not a regular
icosahedron, though. It's probably the "pseudoicosahedron" shown in
Steven Dutch's site above:
$$\href{http://www.uwgb.edu/dutchs/symmetry/isometuc.htm}{\includegraphics[max width=0.65\linewidth]{../images/pseudoicosahedron.png}}$$
Apparently iron pyrite can also form a pseudoicosahedron --- see "Pyrite
No. 7" on the mindat.org website above. Does anyone have actual photos?

To get the corners of a regular icosahedron, we just need to replace the
number $2$ by the golden ratio $\Phi = (\sqrt{5} + 1)/2$:
$$
  \begin{gathered}
    (\Phi,1,0) \quad (\Phi,-1,0) \quad (-\Phi,1,0) \quad (-\Phi,-1,0)
  \\(1,0,\Phi) \quad (-1,0,\Phi) \quad (1,0,-\Phi) \quad (-1,0,-\Phi)
  \\(0,\Phi,1) \quad (0,\Phi,-1) \quad (0,-\Phi,1) \quad (0,-\Phi,-1)  
  \end{gathered}
$$
Now our rectangles are golden rectangles:
$$\href{http://en.wikipedia.org/wiki/Icosahedron#Cartesian_coordinates}{\includegraphics[max width=0.65\linewidth]{../images/Icosahedron-golden-rectangles.png}}$$

Since the pseudoicosahedron does a cheap imitation of this trick, with
the number 2 replacing the golden ratio, the number 2 deserves to be
called the "fool's golden ratio". I thank Carl Brannen for explaining
this out to me!

The regular docahedron is "dual" to the regular icosahedron: the
vertices of the icosahedron are Miller indices for the dodecahedron.
Similarly, I bet the pyritohedron is dual to the pseudoicosahedron.

So, we could call the pyritohedron the "fool's dodecahedron", and the
pseudoicosahedron the "fool's icosahedron". Fool's gold may have
fooled the Greeks into inventing the regular dodecahedron, by giving
them an example of a fool's dodecahedron.

As pointed out by Noam Elkies and James Dolan, there is a sequence of
less and less foolish dodecahedra whose faces have normal vectors
$$
  \begin{gathered}
    (B,A,0) \quad (B,-A,0) \quad (-B,A,0) \quad (-B,-A,0)
  \\(A,0,B) \quad (-A,0,B) \quad (A,0,-B) \quad (-A,0,-B)
  \\(0,B,A) \quad (0,B,-A) \quad (0,-B,A) \quad (0,-B,-A)  
  \end{gathered}
$$
where $A$ and $B$ are the $n$th and $(n+1)$st Fibonacci numbers, respectively.
As $n \to \infty$, these dodecahedra approach a regular dodecahedron in shape,
because the ratio of successive Fibonacci numbers approaches the golden
ratio.

When $A = 1$ and $B = 2$, we get the fool's dodecahedron, since only a fool
would think $2/1$ is the golden ratio.

However, this is not the most foolish of all dodecahedra! The case $A = 1$
and $B = 1$ gives the rhombic dodecahedron, which doesn't even have
pentagonal faces:
$$\href{http://en.wikipedia.org/wiki/Rhombic_dodecahedron}{\includegraphics[max width=0.65\linewidth]{../images/180px-Rhombicdodecahedron.png}}$$

14) Wikipedia, "Rhombic dodecahedron", `http://en.wikipedia.org/wiki/Rhombic_dodecahedron`

So, the rhombic dodecahedron deserves to be called the "moron's
dodecahedron" --- at least for people who think it's actually a regular
dodecahedron.

But actually, even this dodecahedron isn't the dumbest. The Fibonacci
numbers start with $0$:
$$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots$$
So, even more foolish is the case $A = 0$ and $B = 1$. Here our 12 vectors
reduce to just 6 different ones:
$$
  \begin{gathered}
    (1,0,0) \quad (1,-0,0) \quad (-1,0,0) \quad (-1,-0,0)
  \\(0,0,1) \quad (-0,0,1) \quad (0,0,-1) \quad (-0,0,-1)
  \\(0,1,0) \quad (0,1,-0) \quad (0,-1,0) \quad (0,-1,-0)  
  \end{gathered}
$$
These are normal to the faces of a cube. So, the cube deserves to be
called the "half-wit's dodecahedron": it doesn't even have 12 faces,
just 6.

Moving in the direction of increasing wisdom, we can consider the case
$A = 2$, $B = 3$. This gives a dodecahedron which is closer to regular than
the pyritohedron. And, apparently it exists in nature! It shows up as
number 12 in this list of crystals:

15) Ian O. Angell and Moreton Moore, "Projections of cubic crystals", section 4: The diagrams, `http://www.iucr.org/education/pamphlets/12/full-text`

They also call this guy a pyritohedron, so presumably some pyrite forms
these less foolish crystals! You can compare it with the $A = 1$, $B = 2$
case here:

16) Ian O. Angell and Moreton Moore, "Projections of cubic crystals, Graphical index of figures", `http://www.iucr.org/education/pamphlets/12/graphical-index`

The $A = 1$, $B = 2$ pyritohedron is [figure 9](http://www.iucr.org/education/pamphlets/12/graphical-index/figures/individual?show=14180),
while the $A = 2$, $B = 3$ pyritohedron is [figure 12](http://www.iucr.org/education/pamphlets/12/graphical-index/figures/individual?show=14144).
It's noticeably better!

Let me wrap up by mentioning a fancier aspect of the dodecahedron which
has been intriguing me lately. I already mentioned it in
["Week 230"](#week230), but in such a general setting that it may
have whizzed by too fast. Let's slow down a bit and enjoy it.

The rotational symmetries of the dodecahedron form a $60$-element subgroup
of the rotation group $\mathrm{SO}(3)$. So, the "double cover" of the rotational
symmetry group of the dodecahedron is a $120$-element subgroup of $\mathrm{SU}(2)$.
This is called the "binary dodecahedral group". Let's call it $G$.

The group $\mathrm{SU}(2)$ is topologically a $3$-sphere, so $G$ acts as left
translations on this $3$-sphere, and we can use a dodecahedron sitting in
the $3$-sphere as a fundamental domain for this action. This gives the
$120$-cell. The quotient $\mathrm{SU}(2)/G$ is the Poincare homology sphere!

But, we can also think of $G$ as acting on $\mathbb{C}^2$. The quotient $\mathbb{C}^2/G$ is
not smooth: it has an isolated singular coming from the origin in $\mathbb{C}^2$.
But as I mentioned in ["Week 230"](#week230), we can form a
"minimal resolution" of this singularity. This gives a holomorphic map
$$p\colon M \to \mathbb{C}^2/G$$
where $M$ is a complex manifold. If we look at the points in $M$ that map to
the origin in $\mathbb{C}^2/G$, we get a union of 8 Riemann spheres, which
intersect each other in this pattern:
$$
  \begin{tikzpicture}
    \begin{knot}[clip width=7]
      \strand[thick] (0,0)
        to [out=left,in=left,looseness=1.75] (0,1)
        to [out=right,in=right,looseness=1.75] (0,0);
      \strand[thick] (0.75,0)
        to [out=left,in=left,looseness=1.75] (0.75,1)
        to [out=right,in=right,looseness=1.75] (0.75,0);
      \strand[thick] (1.5,0)
        to [out=left,in=left,looseness=1.75] (1.5,1)
        to [out=right,in=right,looseness=1.75] (1.5,0);
      \strand[thick] (2.25,0)
        to [out=left,in=left,looseness=1.75] (2.25,1)
        to [out=right,in=right,looseness=1.75] (2.25,0);
      \strand[thick] (3,0)
        to [out=left,in=left,looseness=1.75] (3,1)
        to [out=right,in=right,looseness=1.75] (3,0);
      \strand[thick] (3.75,0)
        to [out=left,in=left,looseness=1.75] (3.75,1)
        to [out=right,in=right,looseness=1.75] (3.75,0);
      \strand[thick] (4.5,0)
        to [out=left,in=left,looseness=1.75] (4.5,1)
        to [out=right,in=right,looseness=1.75] (4.5,0);
      \flipcrossings{1,3,5,7,9,11}
      \strand[thick] (3,-0.8)
        to [out=left,in=left,looseness=1.75] (3,0.2);
    \end{knot}
    \begin{knot}
      \strand[thick] (3,0.2)
        to [out=right,in=right,looseness=1.75] (3,-0.8);
    \end{knot}
  \end{tikzpicture}
$$
Here I've drawn linked circles to stand for these intersecting spheres,
for a reason soon to be clear. But, already you can see that we've got
8 spheres corresponding to the dots in this diagram:
$$
  \begin{tikzpicture}
    \draw[thick] (0,0) node{$\bullet$} to (1,0) node{$\bullet$} to (2,0) node{$\bullet$} to (3,0) node {$\bullet$} to (4,0) node {$\bullet$} to (5,0) node {$\bullet$} to (6,0) node {$\bullet$};
    \draw[thick] (4,0) to (4,-1) node{$\bullet$};
  \end{tikzpicture}
$$
where the spheres intersect when there's an edge between the
corresponding dots. And, this diagram is the Dynkin diagram for the
exceptional Lie group $\mathrm{E}_8$!

I already mentioned the relation between the $\mathrm{E}_8$ Dynkin diagram and the
Poincare homology sphere in ["Week 164"](#week164), but now maybe
it fits better into a big framework. First, we see that if we take the
unit ball in $\mathbb{C}^2$, and see what points it gives in $\mathbb{C}^2/G$, and then take
the inverse image of these under
$$p\colon M \to \mathbb{C}^2/G,$$
we get a $4$-manifold whose boundary is the Poincare homology $3$-sphere.
So, we have a cobordism from the empty set to the Poincare homology
$3$-sphere! Cobordisms can be described using "surgery on links", and
the link that describes this particular cobordism is:
$$
  \begin{tikzpicture}
    \begin{knot}[clip width=7]
      \strand[thick] (0,0)
        to [out=left,in=left,looseness=1.75] (0,1)
        to [out=right,in=right,looseness=1.75] (0,0);
      \strand[thick] (0.75,0)
        to [out=left,in=left,looseness=1.75] (0.75,1)
        to [out=right,in=right,looseness=1.75] (0.75,0);
      \strand[thick] (1.5,0)
        to [out=left,in=left,looseness=1.75] (1.5,1)
        to [out=right,in=right,looseness=1.75] (1.5,0);
      \strand[thick] (2.25,0)
        to [out=left,in=left,looseness=1.75] (2.25,1)
        to [out=right,in=right,looseness=1.75] (2.25,0);
      \strand[thick] (3,0)
        to [out=left,in=left,looseness=1.75] (3,1)
        to [out=right,in=right,looseness=1.75] (3,0);
      \strand[thick] (3.75,0)
        to [out=left,in=left,looseness=1.75] (3.75,1)
        to [out=right,in=right,looseness=1.75] (3.75,0);
      \strand[thick] (4.5,0)
        to [out=left,in=left,looseness=1.75] (4.5,1)
        to [out=right,in=right,looseness=1.75] (4.5,0);
      \flipcrossings{1,3,5,7,9,11}
      \strand[thick] (3,-0.8)
        to [out=left,in=left,looseness=1.75] (3,0.2);
    \end{knot}
    \begin{knot}
      \strand[thick] (3,0.2)
        to [out=right,in=right,looseness=1.75] (3,-0.8);
    \end{knot}
  \end{tikzpicture}
$$

Second, by the "McKay correspondence" described in
["Week 230"](#week230), all this stuff also works for other
Platonic solids! Namely:

If $G$ is the "binary octahedral group" --- the double cover of the
rotational symmetry group of the octahedron --- then we get a minimal
resolution
$$p\colon M \to \mathbb{C}^2/G$$
which yields, by the same procedure as above, a cobordism from the empty
set to the $3$-manifold $\mathrm{SU}(2)/G$.

This cobordism can be described using surgery on this link:
$$
  \begin{tikzpicture}
    \begin{knot}[clip width=7]
      \strand[thick] (0.75,0)
        to [out=left,in=left,looseness=1.75] (0.75,1)
        to [out=right,in=right,looseness=1.75] (0.75,0);
      \strand[thick] (1.5,0)
        to [out=left,in=left,looseness=1.75] (1.5,1)
        to [out=right,in=right,looseness=1.75] (1.5,0);
      \strand[thick] (2.25,0)
        to [out=left,in=left,looseness=1.75] (2.25,1)
        to [out=right,in=right,looseness=1.75] (2.25,0);
      \strand[thick] (3,0)
        to [out=left,in=left,looseness=1.75] (3,1)
        to [out=right,in=right,looseness=1.75] (3,0);
      \strand[thick] (3.75,0)
        to [out=left,in=left,looseness=1.75] (3.75,1)
        to [out=right,in=right,looseness=1.75] (3.75,0);
      \strand[thick] (4.5,0)
        to [out=left,in=left,looseness=1.75] (4.5,1)
        to [out=right,in=right,looseness=1.75] (4.5,0);
      \flipcrossings{1,3,5,7,9,11}
      \strand[thick] (3,-0.8)
        to [out=left,in=left,looseness=1.75] (3,0.2);
    \end{knot}
    \begin{knot}
      \strand[thick] (3,0.2)
        to [out=right,in=right,looseness=1.75] (3,-0.8);
    \end{knot}
  \end{tikzpicture}
$$
which encodes the Dynkin diagram of $\mathrm{E}_7$:
$$
  \begin{tikzpicture}
    \draw[thick] (0,0) node{$\bullet$} to (1,0) node{$\bullet$} to (2,0) node{$\bullet$} to (3,0) node {$\bullet$} to (4,0) node {$\bullet$} to (5,0) node {$\bullet$};
    \draw[thick] (3,0) to (3,-1) node{$\bullet$};
  \end{tikzpicture}
$$
And, if $G$ is the "binary tetrahedral group" --- the double cover of the
rotational symmetry group of the tetrahedron --- then a minimal resolution
$$p\colon M \to \mathbb{C}^2/G$$
yields, by the same procedure as above, a cobordism from the empty set
to the $3$-manifold $\mathrm{SU}(2)/G$. This cobordism can be described using surgery
on this link:
$$
  \begin{tikzpicture}
    \begin{knot}[clip width=7]
      \strand[thick] (1.5,0)
        to [out=left,in=left,looseness=1.75] (1.5,1)
        to [out=right,in=right,looseness=1.75] (1.5,0);
      \strand[thick] (2.25,0)
        to [out=left,in=left,looseness=1.75] (2.25,1)
        to [out=right,in=right,looseness=1.75] (2.25,0);
      \strand[thick] (3,0)
        to [out=left,in=left,looseness=1.75] (3,1)
        to [out=right,in=right,looseness=1.75] (3,0);
      \strand[thick] (3.75,0)
        to [out=left,in=left,looseness=1.75] (3.75,1)
        to [out=right,in=right,looseness=1.75] (3.75,0);
      \strand[thick] (4.5,0)
        to [out=left,in=left,looseness=1.75] (4.5,1)
        to [out=right,in=right,looseness=1.75] (4.5,0);
      \flipcrossings{1,3,5,7,9,11}
      \strand[thick] (3,-0.8)
        to [out=left,in=left,looseness=1.75] (3,0.2);
    \end{knot}
    \begin{knot}
      \strand[thick] (3,0.2)
        to [out=right,in=right,looseness=1.75] (3,-0.8);
    \end{knot}
  \end{tikzpicture}
$$
which encodes the Dynkin diagram of $\mathrm{E}_6$:
$$
  \begin{tikzpicture}
    \draw[thick] (0,0) node{$\bullet$} to (1,0) node{$\bullet$} to (2,0) node{$\bullet$} to (3,0) node {$\bullet$} to (4,0) node {$\bullet$};
    \draw[thick] (2,0) to (2,-1) node{$\bullet$};
  \end{tikzpicture}
$$

I don't fully understand this stuff, that's for sure. But, I want to.
The Platonic solids are still full of mysteries.

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

**Addenda:** Someone with the handle "Dileffante" has found another
nice example of the dodecahedron in nature --- and even in Nature:

> While perusing a Nature issue I found this short notice on a paper,
> and I remembered that in your talk (which I saw online) you mentioned
> that the dodecahedron was not found in nature. Now I see in
> ["Week 241"](#week241) that there are some things dodecahedral after all, but
> nevertheless, I send this further dodecahedron which was missing
> there.
>
> Nature commented in issue 7075:
>
> 15) "The complete Plato", _Nature_ **439** (26 January 2006), 372--373.
>
> > According to Plato, the heavenly ether and the classical elements ---
> > earth, air, fire and water --- were composed of atoms shaped like
> > polyhedra whose faces are identical, regular polygons. Such shapes
> > are now known as the Platonic solids, of which there are five: the
> > tetrahedron, cube, octahedron, icosahedron and dodecahedron.
> > Microscopic clusters of atoms have already been identified with all
> > of these shapes except the last.
> >
> > Now, researchers led by José Luis Rodríguez-López of the Institute
> > for Scientific and Technological Research of San Luis Potosé in
> > Mexico and Miguel José-Yacamén of the University of Texas, Austin,
> > complete the set. They find that clusters of a gold-palladium alloy
> > about two nanometres across can adopt a dodecahedral shape.
>
> The article is in:
>
> 16) Juan Martín Montejano-Carrizales, José Luis Rodríguez-López, Umapada Pal, Mario Miki-Yoshida and Miguel José-Yacamán, "The completion of the Platonic atomic polyhedra: the dodecahedron", _Small_ **2** (2006), 351--355.
>
> Here's the abstract:
>
> > Binary AuPd nanoparticles in the 1--2 nm size range are synthesized.
> > Through HREM imaging, a dodecahedral atomic growth pattern of five
> > fold axis is identified in the round shaped (85%) particles. Our
> > results demonstrate the first experimental evidence of this Platonic
> > atomic solid at this size range of metallic nanoparticles. Stability
> > of such Platonic structures are validated through theoretical
> > calculations.
>
> Either there is some additional value in the construction, or the
> authors (and Nature editors) were unaware of dodecahedrane.

Dodecahedrane is a molecule built from carbon and hydrogen --- a bit
different from an "atomic cluster" of the sort discussed here. It's a
matter of taste whether that's important, but I bet these
gold-palladium nanoparticles occur in nature, while dodecahedrane seems
to be unstable.

My friend Geoffrey Dixon contributed these pictures of Platonic life
forms:
$$\includegraphics[max width=0.65\linewidth]{../images/platonic_lifeforms.jpg}$$
They look a bit like Ernst Haeckel's pictures from his book
"Kunstformen der Natur" (artforms of nature).

Finally, here's a really important addendum: in March 2009, Lieven le
Bruyn discovered that the ancient Scots did *not* carve stone balls to
look like Platonic solids! The whole story is something between a hoax
and a series of misunderstandings:

17) Lieven le Bruyn, "The Scottish solids hoax", from his blog _neverendingbooks_, March 25, 2009, `http://www.neverendingbooks.org/index.php/the-scottish-solids-hoax.html`

18) John Baez, "Who discovered the icosahedron?", talk at the _Special Session on History and Philosophy of Mathematics_, 2009 Fall Western Section Meeting of the AMS, November 7, 2009. Available at `http://math.ucr.edu/home/baez/icosahedron/`

You can read a bunch of freewheeling discussions triggered by this
Week's Finds at the [$n$-Category Café](http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html).

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

> *The essence of mathematics lies in its freedom.*
> 
> --- Georg Cantor