# October 26, 2008 {#week271}

This week I'll talk about quasicrystals and how they arise from the
interplay between crystallographic and noncrystallographic Coxeter
groups. I'll describe Jeffrey Morton's new paper on groupoids and
2-vector spaces, and Stephen Summers' review of new work on
constructive quantum field theory. But first - more pictures of
Jupiter's moon Io!

Here's a great photo of volcanic activity on Io - the "Masubi plume":

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

1) NASA Photojournal, Masubi plume on Io,
`http://photojournal.jpl.nasa.gov/catalog/PIA02502`

You can see hot gas and dust shooting 100 kilometers up into the
atmosphere!

Here's another:

::: {align="center"}
![](io_pele_eruption.jpg){width="450"}
:::

2) Solarviews, Pele volcano and Pillan Patera,
`http://www.solarviews.com/eng/iopele.htm`

In front we see a volcanic feature called Pillan Patera. Over the
horizon we see an enormous eruption 300 kilometers high coming from the
most intense persistent hot spot on Io: Pele. This seems to be an active
lava lake inside a volcanic depression, or "patera", about 20 x 30
kilometers in size.

But Pillan Patera is no slouch either when it comes to eruptions. Look
at these "before and after" pictures taken 5 months apart in 1997:

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

3) NASA Photojournal, Arizona-sized Io eruption,
`http://photojournal.jpl.nasa.gov/catalog/PIA00744`

The big red ring is sulfur spewed out by Pele. But the exciting new
feature in the "after" picture is the dark blotch centered at Pillan
Patera. It's 400 kilometers in diameter, roughly the size of Arizona.
It consists of about 50 cubic kilometers of lava laid down by a big
eruption. At the peak of the activity, 10,000 cubic meters of lava were
spewing out each second. This was the largest volcanic eruption ever
seen, anywhere!

For more, try these:

4) A.G. Davies et al, Thermal signature, eruption style and eruption
evolution at Pele and Pillan on Io, Jour. Geophys. Research 106 (2001),
33,079-33,103. Also available at
`http://europa.la.asu.edu/pgg/associates/members/williams/gw/pdf/2001Daviesetal.pdf`

5) Jani Radebaugh et al, Observations and temperatures of Io's Pele
Patera from Cassini and Galileo spacecraft images, Icarus 169 (2004),
65-79.

In case you're wondering about the red sulfur around Pele versus the
yellow sulfur you saw last week, let me say a bit about that. Sulfur
comes in an incredible number of forms, or allotropes:

6) Wikipedia, Allotropes of sulfur,
`http://en.wikipedia.org/wiki/Allotropes_of_sulfur`

It can form different molecules consisting of 2 to 20 atoms. The most
common form on Earth is α-sulfur:
[rhombic](http://en.wikipedia.org/wiki/Orthorhombic_crystal_system)
crystals made of ring-shaped molecules containing 8 atoms each:

::: {align="center"}
![](300px-Cyclooctasulfur-above-3D-balls.png)
:::

α-sulfur is lemon yellow, but above 95 degrees Celsius it gradually
turns into pale yellow \beta-sulfur: the ring-shaped molecules reorganize to
form crystals with less symmetry -
[monoclinic](http://en.wikipedia.org/wiki/Monoclinic_crystal_system)
crystals, to be precise.

Sulfur melts around 115 Celsius. But when you heat it above 160 Celsius,
something weird happens: contrary to the usual pattern for liquids, it
gets more viscous as it gets hotter! The reason: the atoms start forming
long chain polymers, called "catena sulfur". As these predominate, the
stuff gets darker in color: first orange, then red, then dark red, and
finally almost black. Blecch! If you then cool it suddenly, it can form
a red amorphous solid. And that, presumably, is what we see in the ring
around Pele. []{#tilings}

Now, from crystals to quasicrystals...

Last week I asked if quasicrystals with approximate 5-fold symmetry
could be obtained by slicing lattices in higher dimensions. Greg Egan
answered - yes! He even has a beautiful Java applet that demonstrates
it:

7) Greg Egan, deBruijn, `http://www.gregegan.net/APPLETS/12/12.html`

It shows some nice quasiperiodic tilings of the plane with approximate
n-fold symmetry, made by cleverly slicing a cubical lattice in
n-dimensional space. Here's a piece of one for n = 4:

::: {align="center"}
![](tiling_4.jpg)
:::

The idea comes from this paper:

8) N. G. deBruijn, Algebraic theory of Penrose's nonperiodic tilings
of the plane, I, II, Nederl. Akad. Wetensch. Indag. Math. 43 (1981),
39-52, 53-66.

When n is odd, we can also get deBruijn's tiling by slicing the A_{n-1}
lattice in $(n-1)$-dimensional space. You're probably most familiar with
the A_3 lattice, which shows up when you stack oranges:

::: {align="center"}
![](close-packed_spheres.jpg)
:::

You'll notice this pattern has tetrahedral symmetry. The symmetry group
of the tetrahedron is also called the A_3 Coxeter group. It's the
group of all permutations of 4 things (the corners of the tetrahedron).
This contains the symmetry group of the square, since that group
contains some but not all permutations of the 4 corners of the square.
Indeed, if you view a regular tetrahedron from the correct angle, it
looks like a square!

This pattern goes on for higher n. Last week I spoke about the A_4
lattice, whose symmetry group consists of all permutations of 5 things -
namely the 5 corners of a $4$-simplex, which is the 4d analogue of a
tetrahedron. I explained how this group contains the symmetry group of
the pentagon. Indeed, if you view a $4$-simplex from the correct angle, it
looks like a pentagon!

So, it's not surprising that we can get a quasiperiodic tiling of the
plane with approximate 5-fold symmetry by taking a 2d slice of the A_4
lattice and doing a few other tricks. Here's a piece:

::: {align="center"}
![](tiling_5.jpg)
:::

But this generalizes: the symmetries of an $(n-1)$-simplex include the
symmetries of a regular $n$-gon. In other words, just as this Coxeter
group, the symmetry group of the pentagon:

       5
     o---o

sits inside the A_4 Coxeter group:

     o---o---o---o

similarly the symmetries of a hexagon:

       6
     o---o  

sit inside the A_5 Coxeter group:

     o---o---o---o---o

and so on: the noncrystallographic Coxeter groups I_2(n) sit nicely
inside the Coxeter groups A_{n+1}. But the really cool part is how
deBruijn uses these to get quasiperiodic tilings of the plane!

You can see details on Greg Egan's page above. Here's a piece of the
tiling for n = 7, again courtesy of Egan's applet:

::: {align="center"}
![](tiling_7.jpg)
:::

And this idea generalizes to the *other* two noncrystallographic Coxeter
groups. Remember, there are just two more:

-   H_3, the symmetry group of the dodecahedron, with 120 elements;
-   H_4, the symmetry group of the 120-cell, with 120 \times 120 elements.

We can get 3d quasicrystals with approximate dodecahedral symmetry by
cleverly slicing the $6$-dimensional D_6 lattice. This is actually
practical, since there really *are* such quasicrystals in nature. And we
can get 4d quasicrystals with approximate 120-cell symmetry by cleverly
slicing the \mathrm{E}_8 lattice! This is just incredibly cool as pure
mathematics:

9) Veit Elser and Neil Sloane, A highly symmetric four-dimensional
quasicrystal, J. Phys. A 20 (1987), 6161-6168. Also available at
`http://akpublic.research.att.com/~njas/doc/Elser.ps`

10) J. F. Sadoc and R. Mosseri, The \mathrm{E}_8 lattice and quasicrystals:
geometry, number theory and quasicrystals, J. Phys. A 26 (1993),
1789-1809.

11) Robert V. Moody and J. Patera, Quasicrystals and icosians, J. Phys.
A. 26 (1993), 2829-2853.

Yes, this is the same Moody who helped invent Kac-Moody algebras! For
the last decade or so he's been working on quasicrystals. In
["Week 20"](#week20) I explained the "icosians" - a subring of
the quaternions built from the symmetries of a dodecahedron - and how
Conway and Sloane used them to construct the \mathrm{E}_8 lattice. Moody's
article uses the icosians to study the 4d quasicrystals that we get by
slicing the \mathrm{E}_8 lattice.

While they may seem remote from the real world, these 4d quasicrystals
can be further sliced to give 3d quasicrytals with approximate
dodecahedral symmetry. So in some sense, the quasicrystals we find in
nature are "shadows" of the \mathrm{E}_8 lattice... trying their best to have
a symmetry that can only exist in 8 dimensions, but never quite
succeeding.

I love this idea, because it's gotten me over my fear of quasicrystals.
They look unruly and complicated, but now I see that some of them have
close ties to the beautiful, perfectly symmetrical world of Dynkin
diagrams. The "noncrystallographic" Coxeter groups are really
"quasicrystallographic"!

Next let me discuss this paper by Jeffrey Morton:

11) Jeffrey Morton, 2-vector spaces and groupoids, available as
[`arXiv:0810.2361`](http://arxiv.org/abs/0810.2361).

It's an important new twist in the Tale of Groupoidification! As part
of this tale, in ["Week 256"](#week256) I described a functor from
the category with

-   finite groupoids as objects,
-   equivalence classes of spans of finite groupoids as morphisms

to the category with

-   finite-dimensional vector spaces as objects,
-   linear operators as morphisms.

I called this "degroupoidification". The idea is that a lot of linear
algebra has an underlying purely combinatorial "skeleton" that
doesn't involve the complex numbers - just symmetry in its purest form.
Groupoidification is quest to strip the fat off linear algebra and do it
using groupoids.

Jeffrey boosts this idea up one notch, getting a 2-functor from the
$2$-category with

-   finite groupoids as objects,
-   spans of finite groupoids as morphisms,
-   equivalence classes of spans of spans of finite groupoids as
    $2$-morphisms

to the $2$-category with

-   finite-dimensional 2-vector spaces as objects,
-   linear functors as morphisms,
-   natural transformations as $2$-morphisms.

Here by "finite-dimensional 2-vector space" I really mean a
"Kapranov-Voevodsky 2-vector space". That's a category equivalent to
Vect^n for some n, where Vect is the category of finite-dimensional
vector spaces. A "linear functor" is one that's linear on each
homset. More concretely, we can describe a linear functor

F\colon Vect^n \to Vect^m^

as an m \times n matrix of finite-dimensional vector spaces, just as we can
describe a linear operator

F\colon C^n \to C^m^

as a m \times n matrix of complex numbers.

This suggests that Jeffrey is secretly talking about a categorified
version of Heisenberg's "matrix mechanics" - and that's true. I want
to explain that. But I'm getting really sick of saying "finite" and
"finite-dimensional". So, henceforth I'll leave out those
adjectives... but they're really always there. Okay?

Degroupoidification turns each groupoid into a vector space... but in
fact it gives more: a Hilbert space! Similarly, Jeffrey's process
actually turns each groupoid into a 2-Hilbert space. I proved that a
long time ago:

12) John Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv.
Math. 127 (1997), 125-189. Also available as
[`arXiv:q-alg/9609018`](http://arXiv.org/abs/arXiv:q-alg/9609018).

So, just as degroupoidification reveals that a fair amount of quantum
mechanics can be done with groupoids instead of vector spaces,
Jeffrey's process reveals that a fair amount of *categorified* quantum
mechanics can also be done with groupoids!

Categorified quantum mechanics becomes important when we go from the
physics of particles (which is really field theory on 1d spacetimes) to
the physics of strings (which is really field theory on 2d spacetimes).
The simplest case is "topological string theory", also known as
"extended 2d topological quantum field theory". And the simplest
example of such a theory is the "Dijkgraaf-Witten model": a gauge
theory with a finite gauge group.

In his thesis:

13) Jeffrey Morton, Extended TQFT's and Quantum Gravity, Ph.D. thesis,
U. C. Riverside, 2007. Available at
[`arXiv:0710.0032`](http://arxiv.org/abs/0710.0032).

Jeffrey showed that a special case, the "untwisted" Dijkgraaf-Witten
model, gives a weak 2-functor from the weak $2$-category with

-   0d manifolds as objects,
-   1d cobordisms between these as morphisms,
-   equivalence classes of 2d cobordisms between these as $2$-morphisms

to the weak $2$-category with

-   finite-dimensional 2-vector spaces as objects,
-   linear functors as morphisms,
-   natural transformations as $2$-morphisms.

Composing this 2-functor with the 2-functor I just described, he gets
the untwisted Dijkgraaf-Witten model as an extended TQFT! And in fact,
he does it in all dimensions, not just dimension 2.

By the way, most of the $2$-categories and 2-functors here are "weak".
Also by the way, Jeffrey constructed the above cobordism $2$-category in
an earlier paper, which I discussed in ["Week 242"](#week242). He
recently polished up this paper, changing the title to make it focus on
the algebraic essence of his construction:

14) Jeffrey Morton, Double bicategories and double cospans, available
as [`arXiv:math/0611930`](http://arxiv.org/abs/math/0611930).

There's a lot more I could say about this, but not a lot more time. So,
let me wrap up with a pointer to Stephen Summers' review of new work on
constructive quantum field theory.

Constructive quantum field theory is the branch of mathematical physics
where you try to rigorously construct examples of quantum field
theories. I did my Ph.D. thesis on this subject under Irving Segal, but
it was too hard for me, and my heart was never really in it, so I soon
fled - first to classical field theory, and then further.

I recently met Stephen Summers at a conference in honor of von Neumann,
and he tried to call me back to my roots. It turns out there's been a
lot of interesting progress in constructive quantum field theory! I'll
probably keep working on topological quantum field theory and other
wimpy subjects - but it's great to hear someone out there is doing the
hard work of getting physically realistic quantum field theories to make
rigorous mathematical sense.

Here's some of what he has to say:

> The development of the tools and techniques of algebraic quantum field
> theory (AQFT) has reached the point where they can be turned upon the
> question of existence of quantum field models. Although the program of
> constructing models via AQFT is still in its infancy and only a few
> researchers are working in the field, already some encouraging
> successes can be displayed. I personally find it stimulating that the
> ideas employed go well beyond the range of the semiclassical ideas
> which were mathematically developed by researchers in constructive
> quantum field theory in the 70's and 80's. There is no appeal to
> Lagrangians, actions and perturbation theory, nor does one "work in
> the Euclidean realm", and one generally avoids a direct construction
> of strictly local quantum field operators (as these either do not
> exist or are prohibitively difficult to construct), preferring to
> construct more physically relevant quantities such as the scattering
> amplitudes and local "observables". Some of the constructed models
> are local and free, some are local and have nontrivial S-matrices, and
> yet others manifest only certain remnants of locality, although these
> remnants suffice to enable the computation of nontrivial two-particle
> S-matrix elements. This includes models with nontrivial scattering in
> four spacetime dimensions.

This is just the beginning of a fascinating review. Check it out:

15) Stephen J. Summers, Constructive AQFT,
`http://www.math.ufl.edu/~sjs/construction.html`

Also check out his big AQFT page, which lists textbooks and many more
references:

16) Stephen J. Summers, Algebraic quantum field theory,
`http://www.math.ufl.edu/~sjs/aqft.html`

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

*During the journey we commonly forget its goal. Almost every profession
is chosen as a means to an end but continued as an end in itself.
Forgetting our objectives is the most frequent act of stupidity.* -
Friedrich Nietzsche