# May 25, 2008 {#week265}

Today I'd like to talk about the Pythagorean pentagram, Bill Schmitt's
work on Hopf algebras in combinatorics, the magnum opus of Aguiar and
Mahajan, and quaternionic analysis. But first, the astronomy picture of
the week.

I seem to be into moons these days: first Saturn's moon Titan in
["Week 263"](#week263), and then Mars' moon Phobos in
["Week 264"](#week264). On the cosmic scale, our Solar System is
like our back yard. It may not be important in the grand scheme of
things, but we should get to know it and learn to take care of it. It's
got lots of cool moons. So this week, let's talk about
[Europa](http://en.wikipedia.org/wiki/Europa_%28moon%29):

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

1) Astronomy Picture of the Day, Gibbous Europa,
`http://antwrp.gsfc.nasa.gov/apod/ap071202.html`

Europa is the fourth biggest moon of Jupiter, the smallest of the four
seen by Galileo. It's 3000 kilometers in diameter, slightly smaller
than our moon, and it zips around Jupiter once every 3.5 of our days,
though it's almost twice as far from Jupiter as our moon is from us.

It looks like a cracked ball of ice, and that's what it is - at least
near the surface.

Indeed, this ancient impact crater looks like a smashed windshield, or a
frozen lake that's been hit with a sledgehammer:

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

2) NASA Photojournal, Ancient impact basin on Europa,
`http://photojournal.jpl.nasa.gov/catalog/PIA00702`

But this crater, called Tyre, is huge: about as big as the island of
Hawaii, 145 kilometers across! (Beware: this picture is a composite of
three photos taken by the Galileo spacecraft in 1997. It's in false
color designed to show off various structures: the original crater, the
later red cracks, and the blue-green ridges.)

The big question is whether there's liquid water beneath the icy
surface... and if so, maybe life? One model of this moon posits a solid
ice crust. Another says there's liquid water too:

::: {align="center"}
[![](europa_models.gif)](http://photojournal.jpl.nasa.gov/catalog/PIA01669)
:::

3) NASA Photojournal, Model of Europa's subsurface structure,
`http://photojournal.jpl.nasa.gov/catalog/PIA01669`

How can we tell? Europa is the *smoothest* of all solid planets and
moons, with lots of cracks and ridges but few remaining craters. This
suggests either an ocean beneath the surface, or at least ice warm
enough to keep convection going. The region called Conamara Chaos looks
like pack ice here on Earth, hinting at liquid water beneath:

::: {align="center"}
[![](europa_chaos.jpg){width="500"}](http://photojournal.jpl.nasa.gov/jpeg/PIA01127.jpg)
:::

4) NASA Photojournal, Europa: ice rafting view,
`http://photojournal.jpl.nasa.gov/catalog/PIA01127`

The bluish white areas have been blanketed with ice dust ejected from
far away when an impact formed a crater called
[Pwyll](http://en.wikipedia.org/wiki/Pwyll_%28crater%29). The reddish
brown regions could contain salts or sulfuric acid - it's hard to find
out using spectroscopy, since there's too much ice.

Another very nice piece of evidence for *salty* liquid water inside
Europa is that the magnetic field of Jupiter induces electric currents
in this moon, which in turn create their own magnetic fields! These
fields were detected when the Galileo probe swooped closest to Europa
back in 2000:

5) M. G. Kivelson, K. K. Khurana, C. T. Russell, M. Volwerk, R.J.
Walker, and C. Zimmer, Galileo magnetometer measurements: a stronger
case for a subsurface ocean at Europa, Science, 289 (2000), 1340-1343.

At the time, Margaret Kivelson, head of the magnetometer project, said:

> I think these findings tell us that there is indeed a layer of liquid
> water beneath Europa's surface. I'm cautious by nature, but this new
> evidence certainly makes the argument for the presence of an ocean far
> more persuasive. Jupiter's magnetic field at Europa's position
> changes direction every 5-1/2 hours. This changing magnetic field can
> drive electrical currents in a conductor, such as an ocean. Those
> currents produce a field similar to Earth's magnetic field, but with
> its magnetic north pole - the location toward which a compass on
> Europa would point - near Europa's equator and constantly moving. In
> fact, it is actually reversing direction entirely every 5-1/2 hours.

A couple weeks ago, another nice piece of evidence was announced:

::: {align="center"}
[![](europa_polar_wandering_schenk.jpg)](http://www.lpi.usra.edu/science/schenk/europaCropCircles/)\
Scale bar is 100 kilometers long.\
Picture by Paul Schenk, Lunar and Planetary Institute.
:::

6) Paul Schenk, Isamu Matsuyama and Francis Nimmo, True polar wander on
Europa from global-scale small-circle depressions, Nature 453 (2008),
368-371.

Paul Schenk, Scars from Europa's polar wandering betray ocean beneath,
[http://www.lpi.usra.edu/science/schenk/europaCropCircles/](http://www.lpi.usra.edu/science/schenk/europaCropCircles)

There are two arc-shaped depressions exactly opposite each other on
Europa, each hundreds of kilometers long and between .3 and 1.5
kilometers deep. According to the above paper, these scars have just the
right shape to be caused the moon's icy shell rotating a quarter turn
relative to the interior! The authors believe this could happen most
easily if it were floating on an ocean.

If Europa has an ocean under its ice, other questions immediately arise.
How thick is the ice and how deep is the ocean? Some guess 15-30
kilometers of ice atop 100 kilometers of liquid. What keeps it warm?
Heating produced by tidal forces may be the best bet - radioactivity
from the core contributes just about 100 billion watts, not nearly
enough:

7) M. N. Ross and G. Schubert, Tidal heating in an internal ocean model
of Europa, Nature 325 (1987), 133-144.

And then for the really big question: could there be *life* on Europa?
Antarctica has an enormous lake called [Lake
Vostok](http://en.wikipedia.org/wiki/Lake_Vostok) buried under 4
kilometers of ice, and when people drilled into it they found all sorts
of bizarre life forms that had never been seen before. So, especially if
Europa had been warmer once, it's conceivable that life might have
formed there and survives to this day. Of course, the surface of Europa
makes Antarctica look downright balmy: it's -160 Celsius at the
equator. And liquid water below could be mixed with sulfuric acid, or
lots of nasty salts...

Nonetheless, some dream of sending a satellite to Europa, perhaps to
impact it at high velocity and see what's inside, or perhaps to land
and melt down through the ice:

8) Leslie Mullen, Hitting Europa hard (interview of Karl Hibbits),
Astrobiology Magazine, May 1, 2006,
`http://www.astrobio.net/news/article1944.html`

But these dreams may not come true anytime soon. In 2005, NASA cancelled
its ambitious plans for the Jupiter Icy Moons Orbiter:

10) Wikipedia, Jupiter Icy Moons Orbiter,
`http://en.wikipedia.org/wiki/Jupiter_Icy_Moons_Orbiter`

The U.S. Congress, the National Academy of Sciences, and the NASA
Advisory Committee have all supported a mission to Europa, but NASA has
still not funded this project:

11) Leonard David, Europa mission: lost in NASA budget, SPACE.com,
February 7, 2006, `http://www.space.com/news/060207_europa_budget.html`

Unfortunately, NASA still spends most of its money on expensive manned
missions - the Buck Rogers approach to space. They think the public
wants the "glamor" of manned missions. So, while they just safely
landed the Phoenix spacecraft on Mars, they're also busy struggling to
fix a toilet in near earth orbit, on the International Space Station.

To study the underground ocean of Europa, our best hope may lie with the
European Space Agency's "Jovian Europa Orbiter", part of a project
called the Jovian Minisat Explorer:

12) ESA Science and Technology, Jovian Minisat Explorer,
`http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=35982`

This hasn't been funded yet, and there's no telling if it ever will.
But people are already working to make sure Europa doesn't get
contaminated by bacteria from Earth:

13) National Research Council, Preventing the Forward Contamination of
Europa, The National Academies Press, Washington, DC, 2000. Also
available at `http://www.nap.edu/catalog.php?record_id=9895`

In fact the US and many other countries are obligated to do this, since
they signed a United Nations treaty that requires it.

The Galileo probe had not been sterilized in a way that would kill
[extremophiles](http://en.wikipedia.org/wiki/Extremophile) - organisms
that survive extreme conditions. So, the National Research Council
recommended that NASA crash Galileo into Jupiter when its mission was
over, to avoid an accidental collision with Europa. So, that's what
they did! After 14 years of collecting data about Jupiter and its moons,
Galileo crashed into Jupiter and burned up in its atmosphere on
September 21, 2004.

Maybe I'll talk about other moons of Jupiter next week... the most
interesting ones besides Europa are volcanic, sulfurous Io and icy
Ganymede, biggest of all.

But now let me turn to the Pythagorean pentagram.

The Pythagoreans - that strange Greek cult of vegetarian mathematicians
- were apparently fascinated by the pentagram. Why? I don't think
there's any textual evidence to help us answer this question, but
luckily there's another way to settle it: unsubstantiated wild guesses!

If you take a pentagram and keep on drawing lines through points that
are already present, you can generate this picture:

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

14) James Dolan, Pythagorean pentagram,
`http://math.ucr.edu/home/baez/pythagorean_pentagram.jpg`

This is just the beginning of an infinite picture packed with
pentagrams. The sizes of these pentagrams are related by various powers
of the golden ratio:

Φ = (1 + √5)/2 = 1.6180339...

In particular, if you run up any arm of the big pentagram you'll see
little pentagrams, alternating blue and green in the above picture, each
1/Φ times as big as the one before.

And if you contemplate these, you can see that:

Φ = 1 + 1/Φ

I could explain how, but I prefer to leave it as a fun little puzzle. If
you get stuck, I'll give you a clue later.

This might have interested the Pythagoreans, since it quickly implies
that

Φ = 1 + 1/Φ

    = 1 + 1/(1 + 1/Φ)

    = 1 + 1/(1 + 1/(1 + 1/Φ))

    = 1 + 1/(1 + 1/(1 + 1/(1 + 1/Φ)))

and so on. This means that the continued fraction expansion of Φ never
ends, so it must be irrational! There's some evidence that early Greeks
were interested in continued fraction expansions... you can read about
that in this marvelous speculative book:

15) David Fowler, The Mathematics Of Plato's Academy: A New
Reconstruction, 2nd edition, Clarendon Press, Oxford, 1999. Review by
Fernando Q. Gouvêa for MAA Online available at
<https://www.maa.org/press/maa-reviews/the-mathematics-of-platos-academy-a-new-reconstruction>

If so, we can imagine that early Greek mathematicians discovered the
irrationality of the golden ratio by contemplating the Pythagorean
pentagram.

I recently gave a talk about this and other fun aspects of the number 5
at George Washington University and Google.

I was invited to Google by my student Mike Stay - more about that some
other day, perhaps. But I'd been invited to George Washington
University by Bill Schmitt. We went to grad school together. While I was
studying quantum field theory with Irving Segal, he was studying
combinatorics with Gian-Carlo Rota. Later he taught me about Joyal's
"especes de structures", also known as "species" or "structure
types". Later still, these turned out to be deeply related to the
quantum harmonic oscillator and Feynman diagrams! For more on that, see
["Week 185"](#week185) and ["Week 202"](#week202).

Bill has always been interested in getting Hopf algebras from structure
types. The idea is implicit in some work of Rota:

16) Saj-Nicole Joni and Gian-Carlo Rota, Coalgebras and bialgebras in
combinatorics, Studies in Applied Mathematics 61 (1979), 93-139.

Gian-Carlo Rota, Hopf algebras in combinatorics, in Gian-Carlo Rota on
Combinatorics: Introductory Papers and Commentaries, ed. J. P. S. Kung,
Birkhauser, Boston, 1995.

but my favorite explanation is here:

17) William R. Schmitt, Hopf algebras of combinatorial structures,
Canadian Journal of Mathematics 45 (1993), 412-428. Also available at
`http://home.gwu.edu/~wschmitt/papers/hacs.pdf`

Let me sketch the simplest result in this paper! For starters, recall
that a structure type is any sort of structure you can put on finite
sets. In other words, it's a functor

F\colon \mathsf{FinSet}_0 \to \mathsf{Set}

where \mathsf{FinSet}_0 is the groupoid of finite sets and bijections. The idea
is that for any finite set X, F(X) is the set all of structures of the
given type that we can put on X. A good example is F(X) = 2^X^, the set
of 2-colorings of X.

Starting from this, we can form a groupoid of F-structured finite sets
and structure-preserving bijections. For example, the groupoid of
2-colored finite sets and color-preserving bijections. The idea should
be obvious, but it's good to make it precise. For category hotshots
it's just the groupoid of "elements" of F, called elt(F). But if
you're not a hotshot yet, I should explain this.

An object of elt(F) is a finite set X together with an element a in
F(X). A morphism of elt(F), say

f\colon (X,a) \to (X',a')

is a bijection

f\colon X \to X'

such that

F(f)(a) = a'

In other words: f is a bijection that carries the F-structure on X to
the F-structure on X'.

Anyway: given a structure type F, we can form a vector space B~F~ whose
basis consists of isomorphism classes of elt(F). And in the paper above,
Bill describes various ways to make B~F~ into various kinds of coalgebra
or Hopf algebra.

I'll only explain the simplest one. There are lots of structure types
where you can "restrict" a structure on a big set to a structure on a
smaller set. For example, a 2-coloring of a set restricts to a
2-coloring of any subset. Let's call such a thing a "structure type
with restriction".

Technically, a structure type with restriction is a functor

F\colon Inj^op^ \to \mathsf{Set}

where Inj is the category of finite sets and injections. When we have
such a thing, the inclusion

i\colon X \to X'

of a little set X in a bigger set X' gives a map

F(i)\colon F(X') \to F(X)

that says how to restrict F-structures on X' to F-structures on X.

In this situation, Bill shows that the vector space B~F~ becomes a
cocommutative coalgebra. In particular, it gets a comultiplication

\Delta\colon B~F~ \to B~F~ \otimes B~F~

which satisfies laws just like the commutative and associative laws for
ordinary multiplication, only "backwards".

The idea is simple: we comultiply a finite set with an F-structure on it
by chopping the set in two parts in all possible ways and using our
ability to restrict the F-structure to each part. I could write down the
formula, but it's better to guess it and then check your guess in
Bill's paper! See his Proposition 3.1.

After Bill came up with this stuff, the connection between Hopf algebras
and combinatorics became a big business - largely due to Kreimer's work
on Hopf algebras and Feynman diagrams. I talked about this back in
["Week 122"](#week122) - but here's a more recent review, with a
hundred references for further study:

18) Kurusch Ebrahimi-Fard and Dirk Kreimer, Hopf algebra approach to
Feynman diagram calculations, available as
[`arXiv:hep-th/0510202`](http://arXiv.org/abs/arXiv:hep-th/0510202).

This yields lots of applications of Bill's ideas to quantum physics. I
have no idea how this huge industry is related to my work with James
Dolan and Jeffrey Morton on structure types, more general "stuff
types", quantum field theory and Feynman diagrams. But, maybe you can
figure it out if you read these:

19) John Baez and Derek Wise, Quantization and Categorification.\
Fall 2003 notes: `http://math.ucr.edu/home/baez/qg-fall2003`\
Winter 2004 notes: `http://math.ucr.edu/home/baez/qg-winter2004/`\
Spring 2004 notes: `http://math.ucr.edu/home/baez/qg-spring2004/`

20) Jeffrey Morton, Categorified algebra and quantum mechanics, Theory
and Applications of Categories 16 (2006), 785-854. Available at
`http://www.emis.de/journals/TAC/volumes/16/29/16-29abs.html` and as
[`arXiv:math/0601458`](http://arxiv.org/abs/math/0601458).

While you're mulling over these ideas, it might pay to ponder this
paper Bill told me about:

21) Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and
Hopf algebras, available at `http://www.math.tamu.edu/~maguiar/a.pdf`

It's 588 pages long! It's a bunch of very sophisticated combinatorics
touching on ideas dear to my heart: q-deformation, species, Fock space,
and higher categories. I can't summarize it, but here are some
immediately gripping portions:

-   Chapter 5, "Higher monoidal categories". Here they discuss
    "n-monoidal categories", which are categories equipped with a list
    of tensor products with lax interchange laws relating each tensor
    product to all the later ones on the list:

    (A \otimes_i B) \otimes_j (A' \otimes_i B') \to (A \otimes_j A') \otimes_i (B \otimes_j B')

    for i < j. These gadgets generalize the "iterated monoidal
    categories" of Balteanu, Fiedorowicz, Schwaenzel, Vogt and also
    Forcey - I gave some references on these back in
    ["Week 209"](#week209). The big difference seems to be that the
    Fiederowicz gang has all the tensor products share the same unit.
    That's great for what they want to do - namely, get a kind of
    category whose nerve is an $n$-fold loop space. But, Aguiar and
    Mahajan study a bunch of examples coming from combinatorics where
    different products have different units! It's really these examples
    that are interesting to me, though the abstract concepts are cool
    too.

-   Chapter 7, "Hopf monoids in species". Here they use "species" to
    mean what I'd call "linear structure types", that is, functors

    F\colon \mathsf{FinSet}_0 \to Vect

    where Vect is the category of vector spaces. In Section 7.9 they
    take Bill Schmitt's trick for getting cocommutative coalgebras from
    structure types with restriction, and use it to get cococommutative
    comonoids in the category of linear structure types! In Section 7.10
    they take another trick to get coalgebras from structure types:

    22) William R. Schmitt, Incidence Hopf algebras, Journal of Pure
    and Applied Algebra 96 (1994), 299-330. Also available at
    `http://home.gwu.edu/~wschmitt/papers/iha.pdf`

    and do something similar with that.

-   Chapter 9, "From species to graded vector spaces: Fock functors".
    This studies what happens when you turn a Hopf monoid in the
    category of linear structure types into a graded Hopf algebra - a
    kind of generalized Fock space.

-   Chapter 11, "Hopf monoids from geometry". Here they get Hopf
    monoids from the A_n Coxeter complexes, using a lot of ideas
    related to Jacques Tits' theory of buildings. There's a lot of
    q-deformation going on here! All these ideas are close to my heart.

You can get more of a sense of what Aguiar is up to by looking at his
homepage. I'll just list a *few* of the cool papers there:

23) Marcelo Aguiar's homepage, `http://www.math.tamu.edu/~maguiar/`

Marcelo Aguiar, Internal categories and quantum groups, Ph.D. thesis,
Cornell University, August 1997. Available at
`http://www.math.tamu.edu/~maguiar/thesis2.pdf`

Marcelo Aguiar, Braids, q-binomials and quantum groups, Advances in
Applied Mathematics 20 (1998) 323-365. Also available at
`http://www.math.tamu.edu/~maguiar/braids.ps.gz`

Marcelo Aguiar and Swapneel Mahajan, Coxeter groups and Hopf algebras,
Fields Institute Monographs, Volume 23, AMS, Providence, RI, 2006. Also
available at `http://www.math.tamu.edu/~maguiar/monograph.pdf`

Check out the mysterious table of "generalized binomial coefficients"
in the second of these papers - it suggests many links between different
subjects of mathematics!

I was going to say a bit about quaternionic analysis, but now I'm worn
out. So, I'll just say that anyone interested in generalizing complex
analysis to the quaternions must read two papers. The first I had
managed to lose for a long time... but now I've found it again:

24) Anthony Sudbery, Quaternionic analysis, Math. Proc. Camb. Φl. Soc.
85 (1979), 199-225. Available at
`http://citeseer.ist.psu.edu/10590.html` and (slightly different
version)
`http://theworld.com/~sweetser/quaternions/ps/Quaternionic-analysis.pdf`

The second was brought to my attention by David Corfield:

25) Igor Frenkel and Matvei Libine, Quaternionic analysis,
representation theory and physics, available as
[`arXiv:0711.2699`](http://arxiv.org/abs/0711.2699)

Since Igor Frenkel is a bigshot, this paper may finally bring this
neglected subject some of the attention it deserves! Like Corfield,
I'll just quote the abstract, to make your mouth water:

> We develop quaternionic analysis using as a guiding principle
> representation theory of various real forms of the conformal group. We
> first review the Cauchy-Fueter and Poisson formulas and explain their
> representation theoretic meaning. The requirement of unitarity of
> representations leads us to the extensions of these formulas in
> Minkowski space, which can be viewed as another real form of
> quaternions. Representation theory also suggests a quaternionic
> version of the Cauchy formula for the second order pole. Remarkably,
> the derivative appearing in the complex case is replaced by the
> Maxwell equations in the quaternionic counterpart. We also uncover the
> connection between quaternionic analysis and various structures in
> quantum mechanics and quantum field theory, such as the spectrum of
> the hydrogen atom, polarization of vacuum, and one-loop Feynman
> integrals. We also make some further conjectures. The main goal of
> this and our subsequent paper is to revive quaternionic analysis and
> to show profound relations between quaternionic analysis,
> representation theory and four-dimensional physics.

Finally, here's a clue for the Pythagorean pentagram puzzle. To prove
that

Φ = 1 + 1/Φ,

show the length of the longest red interval here is the sum of the
lengths of the two shorter ones:

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

26) James Dolan and John Baez, annotated picture of Pythagorean
pentagram, `http://math.ucr.edu/home/baez/golden_ratio_pentagram.jpg`

For more on the golden ratio, try ["Week 203"](#week203). For more
on its relation to the dodecahedron, see ["Week 241"](#week241).

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

**Addenda:** Here's another stunning picture of the ridges and cracks
on Europa:

::: {align="center"}
[![](europa_ice_cracks.jpg)](http://photojournal.jpl.nasa.gov/catalog/PIA03002)
:::

27) NASA Photojournal, Blocks in the Europan crust provide more
evidence of subterranean ocean,
`http://photojournal.jpl.nasa.gov/catalog/PIA03002`

You can see more discussion of this Week's Finds at the [$n$-Category Café](http://golem.ph.utexas.edu/category/2008/05/this_weeks_finds_in_mathematic_26.html).
You can also see a list of questions I'd like your help with!

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

*There is geometry in the humming of the strings, there is music in the
spacing of the spheres.* - Pythagoras