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 "week263", and then Mars' moon Phobos in "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:
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:
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:
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:
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. 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:
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/
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 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 - 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:
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 "week185" and "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: FinSet0 → Set
where FinSet0 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) = 2X, 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: (X,a) → (X',a')
is a bijection
f: X → X'
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 BF whose basis consists of isomorphism classes of elt(F). And in the paper above, Bill describes various ways to make BF 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: Injop → Set
where Inj is the category of finite sets and injections. When we have such a thing, the inclusion
i: X → X'
of a little set X in a bigger set X' gives a map
F(i): F(X') → 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 BF becomes a cocommutative coalgebra. In particular, it gets a comultiplication
Δ: BF → BF ⊗ BF
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 "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.
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.
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:
(A ⊗i B) ⊗j (A' ⊗i B') → (A ⊗j A') ⊗i (B ⊗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 "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.
F: FinSet0 → 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.
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
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:
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 "week203". For more on its relation to the dodecahedron, see "week241".
Addenda: Here's another stunning picture of the ridges and cracks on Europa:
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. 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
© 2008 John Baez