- #1
John Baez
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
http://www.ams.org/notices/200606/comm-grunbaum.pdf
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/
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,
http://aimath.org/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,
http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html
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
http://math.ucr.edu/home/baez/octonions/conway_smith/
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
http://www.math.lsa.umich.edu/~jrs/data/coxplanes/
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
http://www.math.lsa.umich.edu/~jrs/data/coxplanes/
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
http://www.math.lsa.umich.edu/~jrs/data/coxplanes/
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
http://www.math.lsa.umich.edu/~jrs/data/coxplanes/
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:
S
/ \
/ \
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 let's us turn spans of sets into matrices of natural numbers.
Given a span of finite sets:
S
/ \
/ \
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 let's 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
S
/ \
/ \
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
math.QA/0601458.
14) Simon Byrne, On Groupoids and Stuff, honors thesis, Macquarie
University, 2005, available at
http://www.maths.mq.edu.au/~street/ByrneHons.pdf and
http://math.ucr.edu/home/baez/qg-spring2004/ByrneHons.pdf
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
groupoids!
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
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
http://philsci-archive.pitt.edu/archive/00001606/
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
quant-ph/0404040.
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,
though.
-----------------------------------------------------------------------
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
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.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
http://www.ams.org/notices/200606/comm-grunbaum.pdf
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/
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,
http://aimath.org/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,
http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html
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
http://math.ucr.edu/home/baez/octonions/conway_smith/
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
http://www.math.lsa.umich.edu/~jrs/data/coxplanes/
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
http://www.math.lsa.umich.edu/~jrs/data/coxplanes/
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
http://www.math.lsa.umich.edu/~jrs/data/coxplanes/
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
http://www.math.lsa.umich.edu/~jrs/data/coxplanes/
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:
S
/ \
/ \
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 let's us turn spans of sets into matrices of natural numbers.
Given a span of finite sets:
S
/ \
/ \
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 let's 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
S
/ \
/ \
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
math.QA/0601458.
14) Simon Byrne, On Groupoids and Stuff, honors thesis, Macquarie
University, 2005, available at
http://www.maths.mq.edu.au/~street/ByrneHons.pdf and
http://math.ucr.edu/home/baez/qg-spring2004/ByrneHons.pdf
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
groupoids!
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
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
http://philsci-archive.pitt.edu/archive/00001606/
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
quant-ph/0404040.
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,
though.
-----------------------------------------------------------------------
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
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
Last edited by a moderator: