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This Week's Finds in Mathematical Physics (Week 247)

  1. Mar 24, 2007 #1
    Also available as http://math.ucr.edu/home/baez/week247.html

    March 23, 2007
    This Week's Finds in Mathematical Physics (Week 247)
    John Baez

    Symmetry has fascinated us throughout the ages. Greek settlers
    in Sicily may have seen irregular 12-sided crystals of pyrite in Sicily
    and dreamt up the regular dodecahedron simply because it was more
    beautiful, more symmetrical. The Alhambra, a Moorish palace in Granada
    built around 1300, has tile patterns with at least 13 of the 17 possible
    symmetry groups:

    1) Branko Gruenbaum, What symmetry groups are present in the Alhambra?,
    Notices of the AMS, 53 (2006), 670-673. Also available at

    You can see some of these patterns here:

    2) Moresque tiles, http://www.spsu.edu/math/tile/grammar/moor.htm

    Recently, Peter Lu and Paul Steinhardt discovered that Islamic tile
    designs include "quasicrystals", patterns that seem to struggle for
    a 5-fold symmetry but never quite reach it, unless we think of them
    as slices of a higher-dimensional lattice:

    3) Peter J. Lu and Paul J. Steinhardt, Decagonal and quasi-crystalline
    tilings in medieval Islamic architecture, Science 315 (2007), 1106-1110.
    Also available at http://www.physics.harvard.edu/~plu/publications/ [Broken]

    Here's one from the I'timad al-Daula mausoleum in the Indian city of
    Agra, built by Islamic conquerors in 1622 - together with a more
    mathematical version constructed by Lu and Steinhardt:

    Here's another, from the Darb-i Imam shrine in Isfahan, Iran, also built
    in the 1600s:

    This came as a big surprise, since everyone had *thought* that the
    math behind quasicrystals was first discovered by Penrose around 1974,
    then seen in nature by Shechtman, Blech, Gratias and Cahn in 1983.
    It goes to show that the appeal of symmetry, even in its subtler forms,
    is very old!

    For more on quasicrystals, try this:

    4) Steven Webber, Quasicrystals, http://www.jcrystal.com/steffenweber/

    Of course, the appeal of symmetry didn't end with ancient Greeks or
    medieval Islamic monarchs. It also seems to have gotten ahold of John
    Fry, chief executive of Fry's Electronics - a chain of retail shops
    whose motto is "Your best buys are always at Fry's". In 1994 he
    set up something called the American Institute of Mathematics. The
    headquarters was in a Fry's store in Palo Alto - not very romantic.
    But last year, this institute announced plans to move to a full-scale
    replica of the Alhambra!

    5) Associated Press, Silicon valley will get Alhambra-like castle,
    August 18, 2006. Available at http://www.msnbc.msn.com/id/14412387/

    And this week, the institute flexed its mighty PR muscles and coaxed
    reporters from the New York Times, BBC, Le Monde, Scientific American,
    Science News, and so on to write about a highly esoteric advance in
    our understanding of symmetry - a gargantuan calculation involving the
    Lie group E8:

    6) American Institute of Mathematics, Mathematicians map E8,

    The calculation is indeed huge. The *answer* takes up 60 gigabytes of
    data - the equivalent of 45 days of music in MP3 format. If this
    information were written out on paper, it would cover Manhattan!

    But what's the calculation *about*? It almost seems a good explanation
    of that would *also* cover Manhattan. I took a stab at it here:

    7) John Baez, News about E8,

    but I only got as far as sketching a description of E8 and some gadgets
    called R-polynomials. Then come Kazhdan-Lusztig polynomials, and
    Kazhdan-Lusztig-Vogan polynomials.... For more details, follow
    the links, especially to the page written by Jeffrey Adams, who led
    the project.

    In weeks to come, I'll say more about some topics tangentially related
    to this calculation - especially flag varieties, representation theory
    and the Weil conjectures. I may even talk about Kazhdan-Lusztig polynomials!

    For starters, though, let's just look at some pretty pictures by
    John Dembridge that hint at the majesty of E8. Then I'll sketch the
    real subject of Weeks to come: symmetry, geometry, and "groupoidification".

    To warm up to E8, let's first take a look at D4, D5, E6, and E7.

    In "week91" I spoke about the D4 lattice. To get this, first take
    a bunch of equal-sized spheres in 4 dimensions. Stack them in a hypercubical
    pattern, so their centers lie at the points with integer coordinates.
    A bit surprisingly, there's a lot of room left over - enough to fit
    in another copy of this whole pattern: a bunch of spheres whose centers
    lie at the points with *half-integer* coordinates!

    If you stick in these extra spheres, you get the densest known packing
    of spheres in 4 dimensions. Their centers form the "D4 lattice".
    It's an easy exercise to check that each sphere touches 24 others.
    The centers of these 24 are the vertices of a marvelous shape called
    the "24-cell" - one of the six 4-dimensional Platonic solids. It looks
    like this:

    8) John Baez, picture of 24-cell, in a review of On Quaternions and
    Octonions: Their Geometry, Arithmetic and Symmetry, by John H. Conway
    and Derek A. Smith, available at

    Here I'm using a severe form of perspective to project 4 dimensions down
    to 2. The coordinate axes are drawn as dashed lines; the solid lines are
    the edges of the 24-cell.

    How about in 5 dimensions? Here the densest known packing of spheres
    uses the "D5 lattice". This is a lot like the D4 lattice... but only
    if you think about it the right way.

    Imagine a 4-dimensional checkerboard with "squares" - really hypercubes! -
    alternately colored red and black. Put a dot in the middle of each
    black square. Voila! You get a rescaled version of the D4 lattice.
    It's not instantly obvious that this matches my previous description,
    but it's true.

    If you do the same thing with a 5-dimensional checkerboard, you get
    the "D5 lattice", by definition. This gives the densest known
    packing of spheres in 5 dimensions. In this packing, each sphere
    has 40 nearest neighbors. The centers of these nearest neighbors
    are the vertices of a solid that looks like this:

    8) John Stembridge, D5 root system, available at

    If you do the same thing with a 6-dimensional checkerboard, you get
    the "D6 lattice"... and so on.

    However, in 8 dimensions something cool happens. If you pack spheres
    in the pattern of the D8 lattice, there's enough room left to stick in an
    extra copy of this whole pattern! The result is called the "E8 lattice".
    It's twice as dense as the D8 lattice.

    If you then take a well-chosen 7-dimensional slice through the origin of
    the E8 lattice, you get the E7 lattice. And if you take a well-chosen
    6-dimensional slice of this, you get the E6 lattice. For precise details
    on what I mean by "well-chosen", see "week65".

    E6 and E7 give denser packings of spheres than D6 and D7. In fact,
    they give the densest known packings of spheres in 6 and 7 dimensions!

    In the E6 lattice, each sphere has 72 nearest neighbors. They form
    the vertices of a solid that looks like this:

    8) John Stembridge, E6 root system, available at

    In the E7 lattice, each sphere has 126 nearest neighbors. They form
    the vertices of a solid like this:

    9) John Stembridge, E7 root system, available at

    In the E8 lattice, each sphere has 240 nearest neighbors. They form
    the vertices of a solid like this:

    10) John Stembridge, E8 root system, available at

    Faithful readers will know I've discussed these lattices often before.
    For how they give rise to Lie groups, see "week63". For more about
    "ADE classifications", see "week64" and "week230". I haven't really
    added much this time, except Stembridge's nice pictures. I'm really
    just trying to get you in the mood for a big adventure involving all
    these ideas: the Tale of Groupoidification!

    If we let this story lead us where it wants to go, we'll meet and
    all sorts of famous and fascinating creatures, such as:

    Coxeter groups, buildings, and the quantization of logic
    Hecke algebras and Hecke operators
    categorified quantum groups and Khovanov homology
    Kleinian singularities and the McKay correspondence
    quiver representations and Hall algebras
    intersection cohomology, perverse sheaves and Kazhdan-Lusztig theory

    However, the charm of the tale is how many of these ideas are unified
    and made simpler thanks to a big, simple idea: groupoidification.

    So, what's groupoidification? It's a method of exposing the combinatorial
    underpinnings of linear algebra - the hard bones of set theory underlying
    the flexibility of the continuum.

    Linear algebra is all about vector spaces and linear maps. One of the
    lessons that gets drummed into you when you study this subject is that
    it's good to avoid picking bases for your vector spaces until you need
    them. It's good to keep the freedom to do coordinate transformations...
    and not just keep it in reserve, but keep it *manifest*!

    As Hermann Weyl wrote, "The introduction of a coordinate system to geometry
    is an act of violence".

    This is a deep truth, which hits many physicists when they study special
    and general relativity. However, as Niels Bohr quipped, a deep truth is one
    whose opposite is also a deep truth. There are some situations where
    a vector space comes equipped with a god-given basis. Then it's foolish
    not to pay attention to this fact!

    The most obvious example is when our vector space has been *defined*
    to consist of formal linear combinations of the elements of some set.
    Then this set is our basis.

    This often happens when we use linear algebra to study combinatorics.

    But if sets give vector spaces, what gives linear operators? Your
    first guess might be *functions*. And indeed, functions between sets
    do give linear operators between their vector spaces. For example,
    suppose we have a function

    f: {livecat, deadcat} -> {livecat, deadcat}

    which "makes sure the cat is dead":

    f(livecat) = deadcat

    f(deadcat) = deadcat

    Then, we can extend f to a linear operator defined on formal
    linear combinations of cats:

    F(a livecat + b deadcat) = a deadcat + b deadcat

    Written as a matrix in the {livecat, deadcat} basis, this looks like

    0 0

    1 1

    (The relation to quantum mechanics here is just a vague hint of
    themes to come. I've deliberately picked an example where the linear
    operator is *not* unitary.)

    So, we get some linear operators from functions... but not all!
    We only get operators whose matrices have exactly a single 1 in
    each column, the rest of the entries being 0. That's because a
    function f: X -> Y sends each element of X to a single element of Y.

    This is very limiting. We can do better if we get operators from
    *relations* between sets. In a relation between sets X and Y,
    an element of X can be related to any number of elements of Y, and
    vice versa. For example, let the relation

    R: {1,2,3,4} -> {1,2,3,4}

    be "is a divisor of". Then 1 is a divisor of everybody, 2 is a
    divisor of itself and 4, 3 is only a divisor of itself, and 4 is
    only a divisor of itself. We can encode this in a matrix:

    1 0 0 0
    1 1 0 0
    1 0 1 0
    1 1 0 1

    where 1 means "is a divisor of" and 0 means "is not a divisor of".

    We can get any matrix of 0's and 1's this way. Relations are really
    just matrices of truth values. We're thinking of them as matrices of
    numbers. Unfortunately we're still far from getting *all* matrices
    of numbers!

    We can do better if we get matrices from *spans* of sets. A span of
    sets, written S: X -/-> Y, is just a set S equipped with functions to
    X and Y. We can draw it like this:

    / \
    / \
    f/ \g
    / \
    v v
    X Y

    This is my wretched ASCII attempt to draw two arrows coming down from
    the set S to the sets X and Y. It's supposed to look like a bridge -
    hence the term "span".

    Spans of sets are like relations, but where you can be related to
    someone more than once!

    For example, X could be the set of Frenchman and Y could be the set of
    Englishwomen. S could be the set of Russians. As you know, every
    Russian has exactly one favorite Frenchman and one favorite
    Englishman. So, f could be the function "your favorite Frenchman",
    and g could be "your favorite Englishman".

    Then, given a Frenchman x and an Englishwoman y, they're related by
    the Russian s whenever s has x as their favorite Frenchman and y as
    their favorite Englishwoman:

    f(s) = x and g(s) = y.

    Some pairs (x,y) will be related by no Russians, others will be related
    by one, and others will be related by more than one! I bet the pair

    (x,y) = (Gerard Depardieu, Emma Thompson)

    is related by at least 57 Russians.

    This idea lets us turn spans of sets into matrices of natural numbers.
    Given a span of finite sets:

    / \
    / \
    f/ \g
    / \
    v v
    X Y

    we get an X x Y matrix whose (x,y) entry is the number of Russians -
    I mean elements s of S - such that

    f(s) = x and g(s) = y.

    We can get any finite-sized matrix of natural numbers this way.

    Even better, there's a way to "compose" spans that nicely matches the
    usual way of multiplying matrices. You can figure this out yourself if
    you solve this puzzle:

    Let X be the set of people on Earth. Let T be the X x X matrix
    corresponding to the relation "is the father of". Why does the matrix
    T^2 correspond to the relation "is the grandfather of"? Let S
    correspond to the relation "is a friend of". Why doesn't the matrix
    matrix S^2 correspond to the relation "is a friend of a friend of"?
    What span does this matrix correspond to?

    To go further, we need to consider spans, not of sets, but of groupoids!

    I'll say more about this later - I suspect you're getting tired.
    But for now, briefly: a groupoid is a category with inverses. Any
    group gives an example, but groupoids are more general - they're the
    modern way of thinking about symmetry.

    There's a way to define the cardinality of a finite groupoid:

    12) John Baez and James Dolan, From finite sets to Feynman diagrams,
    in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Bjorn Engquist
    and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50. Also available
    as math.QA/0004133.

    And, this can equal any nonnegative *rational* number! This lets us
    generalize what we've done from finite sets to finite groupoids, and
    get rational numbers into the game.

    A span of groupoids is a diagram

    / \
    / \
    f/ \g
    / \
    v v
    X Y

    where X, Y, S are groupoids and f, g are functors. If all the groupoids
    are finite, we can turn this span into a finite-sized matrix of nonnegative
    rational numbers, by copying what we did for spans of finite sets.

    There's also a way of composing spans of groupoids, which corresponds
    to multiplying matrices. For details, see:

    13) Jeffrey Morton, Categorified algebra and quantum mechanics, to
    appear in Theory and Application of Categories. Also available as

    14) Simon Byrne, On Groupoids and Stuff, honors thesis, Macquarie
    University, 2005, available at
    http://www.maths.mq.edu.au/~street/ByrneHons.pdf and

    And, the idea of "groupoidification" is that in many cases where
    mathematicians think they're playing around with linear operators
    between vector spaces, they're *actually* playing around with spans of

    This is especially true in math related to simple Lie groups, their
    Lie algebras, quantum groups and the like. While people usually study
    these gadgets using linear algebra, there's a lot of combinatorics
    involved - and where combinatorics and symmetry show up, one invariably
    finds groupoids.

    As the name suggests, groupoidification is akin to categorification.
    But, it's a bit different. In categorification, we try to boost up
    mathematical ideas this way:

    sets -> categories
    functions -> functors

    In groupoidification, we try this:

    vector spaces -> groupoids
    linear operators -> spans of groupoids

    Actually, it's "decategorification" and "degroupoidification" that are
    systematic processes. These processes lose information, so there's no
    systematic way to reverse them. But, as I explained in "week99", it's
    still fun to try! If we succeed, we discover an extra layer of structure
    beneath the math we thought we understood... and this usually makes that
    math *clearer* and *less technical*, because we're not seeing it through
    a blurry, information-losing lens.

    Okay, that's enough for now. On a completely different note, here's
    a book on "structural realism" and quantum mechanics:

    15) Dean Rickles, Steven French, and Juha Saatsi, The Structural
    Foundations of Quantum Gravity, Oxford University Press, Oxford,
    2006. Containing:

    Dean Rickles and Steven French, Quantum gravity meets structuralism:
    interweaving relations in the foundations of physics. Also available at
    http://fds.oup.com/www.oup.co.uk/pdf/0-19-926969-6.pdf [Broken]

    Tian Yu Cao, Structural realism and quantum gravity.

    John Stachel, Structure, individuality, and quantum gravity.
    Also available as gr-qc/0507078.

    Oliver Pooley, Points, particles, and structural realism. Also
    available at http://philsci-archive.pitt.edu/archive/00002939/

    Mauro Dorato and Massimo Pauri, Holism and structuralism in
    classical and quantum general relativity. Also available at

    Dean Rickles, Time and structure in canonical gravity. Also
    available at http://philsci-archive.pitt.edu/archive/00001845/

    Lee Smolin, The case for background independence. Also available
    as hep-th/0507235.

    John Baez, Quantum quandaries: A category-theoretic perspective.
    Also available at http://math.ucr.edu/home/baez/quantum/ and as

    Very loosely speaking - I ain't no philosopher - structural realism is
    the idea that what's "real" about mathematics, or the abstractions in
    physical theories, are not individual entities but the structures, or
    patterns, they form. So, instead of asking tired questions like "What
    is the number 2, really?" or "Do points of spacetime really exist?",
    we should ask more global questions about the roles that structures
    like "natural numbers" or "spacetime" play in math and physics. It's
    a bit like how in category theory, we can only understand an object in
    the context of the category it inhabits.

    Finally, here's a puzzle for lattice and Lie group fans. The dots
    in Stembridge's pictures are the shortest nonzero vectors in the D5,
    E6, E7, and E8 lattices - or in technical terms, the "roots". Of
    course, only for ADE Dynkin diagrams are the roots all of equal length -
    but those are the kind we have here. Anyway: in the D5 case, only 32
    of the 40 roots are visible. The other 8 are hidden in back somewhere.
    Where are they?

    I asked John Stembridge about this and he gave a useful clue. His
    planar pictures show projections of the roots into what he calls the
    "Coxeter plane".

    Recall from "week62" that the "Coxeter group" associated to a Dynkin
    diagram acts as rotation/reflection symmetries of the roots; it's
    generated by reflections through the roots. There's a basis of roots
    called "simple roots", one for each dot in our Dynkin diagram, and
    the product of reflections through all these simple roots is called
    the "Coxeter element" of our Coxeter group - it's well-defined up to
    conjugation. The "Coxeter plane" is the canonical plane on which the
    Coxeter element acts as a rotation.

    A rotation by how much? The order of the Coxeter element is called
    the "Coxeter number" and denoted h, so the Coxeter element acts on the
    Coxeter plane as a rotation of 2pi/h. The Coxeter number is important
    for other reasons, too! Here's how it goes:

    Coxeter group Coxeter number
    A_n n+1
    B_n 2n
    C_n 2n
    D_n 2n-2
    E6 12
    E7 18
    E8 30
    F4 12
    G2 6

    For D5 the Coxeter number is 8, which accounts for the 8-fold symmetry
    of Stembridge's picture in that case. The E8 picture has 30-fold symmetry!
    My D4 picture has 8-fold symmetry, so I must not have been projecting
    down to the Coxeter plane.

    Anyway, this stuff should help answer my puzzle. I don't know the answer,


    Quote of the Week:

    The true spirit of delight, the exaltation, the sense of being more
    than Man, which is the touchstone of the highest excellence, is to
    be found in mathematics as surely as poetry. - Bertrand Russell

    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

    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Apr 8, 2007 #2
    John Baez wrote:

    > Finally, here's a puzzle for lattice and Lie group fans. The dots
    > in Stembridge's pictures are the shortest nonzero vectors in the D5,
    > E6, E7, and E8 lattices - or in technical terms, the "roots". Of
    > course, only for ADE Dynkin diagrams are the roots all of equal length -
    > but those are the kind we have here. Anyway: in the D5 case, only 32
    > of the 40 roots are visible. The other 8 are hidden in back somewhere.
    > Where are they?

    I've made an animated GIF of the D5 lattice at


    You can see a projection of the lattice as it rotates. At one
    t=0, the projection plane is the Coxeter plane, and you can see
    8 of the 40 vertices dissapear behind others. This was the point of John
    baez's puzzle.
    But at another time, even more points disappear, leaving only
    24 points. So we could ask: In which projection do you see the least

    The case of D3 is a bit easier to visualize: It is a cuboctahedron.
    The Coxeter plane is parallel to one of the square faces. If you
    look at the cuboctahedron through this plane, you see 8 of its
    12 points, the bottom square being hidden behind the top one.
    If you look in a suitable plane, you see only 6 vertices.
    The D5 case is a higher dimensional analogue of this.

    Here are some details of how I did the calculation:

    The D5 lattice has 5 roots, which can be arranged in a Dynkin diagram:

    ( 1) ( 0) ( 0) ( 0)
    (-1) (-1) ( 0) ( 0)
    ( 0) ( 1) ( 1) ( 0)
    ( 0) ( 0) (-1) (-1)
    ( 0) ( 0) ( 0) ( 1)
    o (-1)
    ( 0)
    ( 0)
    ( 0)

    The roots represent reflections "through" these directions. You can
    compute the reflection matrix (R_ij) from a root vector (x_i)
    by R_ij = Identity_ij - 2 x_i x_j.
    [The roots are also a basis for the lattice. I don't know if
    all root systems are always also a basis for the lattice]
    Then you multiply the 5 matrices, to get the "Coxeter element".
    This turns out to be a rotation. I don't quite understand the
    significance of this, but I can find the plane of rotation
    by calculating the eigenvectors of the Coxeter element.

    Next we have to find the 40 vertices and 120 edges of the
    polytope. The vertices of the polytope are the points a
    5-sphere touches its neighbours in a D5 sphere packing.
    So they are the 40 points closest to the origin that
    satisfy the D5 criterion: 5 integers whose sum is even.
    These points are just permutations of (+-1, +-1, 0, 0, 0).
    The edges are drawn whenever the distance between 2 points
    is 2, which is the closest between any 2 points in the

    This completes the job: we have vertices, edges, and a
    projection plane. The animation includes a rotation,
    such that alfa=0 corresponds to the Coxeter plane.

    I might make a similar animated GIF of the E8 lattice...

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