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This Week's Finds in Mathematical Physics (Week 265)

  1. May 27, 2008 #1
    Also available at http://math.ucr.edu/home/baez/week265.html

    May 25, 2008
    This Week's Finds in Mathematical Physics (Week 265)
    John Baez

    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,

    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 orbits 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

    2) NASA Planetary Photojournal, Ancient impact basin on Europa,

    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 Planetary Photojournal, Model of Europa's subsurface structure,

    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 Planetary Photojournal, Europa: ice rafting view,

    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),

    Paul Schenk, Scars from Europa's polar wandering betray ocean beneath,

    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 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,

    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,

    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

    Though NASA just safely landed the Phoenix spacecraft on Mars, which
    is wonderful, they still spend tons of money on showy, expensive
    manned missions - the Buck Rogers approach to space. So, 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,

    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 the volcanic, sulfurous
    Io and the 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,

    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:

    Phi = (sqrt(5) + 1)/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/Phi times as big as the one before.

    And if you contemplate these, you can see that:

    Phi = 1 + 1/Phi

    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

    Phi = 1 + 1/Phi

    = 1 + 1/(1 + 1/Phi)

    = 1 + 1/(1 + 1/(1 + 1/Phi))

    = 1 + 1/(1 + 1/(1 + 1/(1 + 1/Phi)))

    and so on. This means that the continued fraction expansion of
    Phi 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

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

    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: FinSet_0 -> Set

    where 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: (X,a) -> (X',a')

    is a bijection

    f: X -> 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: Inj^{op} -> 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 B_F becomes
    a cocommutative coalgebra. In particular, it gets a comultiplication

    Delta: B_F -> B_F tensor 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
    "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:

    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 tensor_i B) tensor_j (A' tensor_i B') ->
    (A tensor_j A') tensor_i (B tensor_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.

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

    F: FinSet_0 -> 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

    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

    Marcelo Aguiar, Braids, q-binomials and quantum groups, Advances in
    Applied Mathematics 20 (1998) 323-365. Also available at

    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

    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. Phil.
    Soc. 85 (1979), 199-225. Available at
    http://citeseer.ist.psu.edu/10590.html and (slightly different version)

    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

    Phi = 1 + 1/Phi,

    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".


    Quote of the Week:

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

    Previous issues of "This Week's Finds" and other expository articles on
    mathematics and physics, as well as some of my research papers, can be
    obtained at


    For a table of contents of all the issues of This Week's Finds, try


    A simple jumping-off point to the old issues is available at


    If you just want the latest issue, go to

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