Also available at http://math.ucr.edu/home/baez/week145.html
February 9, 2000
This Week's Finds in Mathematical Physics (Week 145)
John Baez
I know I promised to talk about homotopy theory and ncategories, but
I've gotten sidetracked into thinking about projective planes, so I'll
talk about that this Week and go back to the other stuff later. Sorry,
but if I don't talk about what intrigues me at the instant I'm writing
this stuff, I can't get up the energy to write it.
So:
There are many kinds of geometry. After Euclidean geometry, one of the
first to become popular was projective geometry. Projective geometry
is the geometry of perspective. If you draw a picture on a piece of
paper and view it from a slant, distances and angles in the picture will
get messed up  but lines will still look like lines. This kind of
transformation is called a "projective transformation". Projective
geometry is the study of those aspects of geometry that are preserved
by projective transformations.
Interestingly, 2dimensional projective geometry has some curious
features that don't show up in higher dimensions. To explain this,
I need to tell you about projective planes.
I talked a bit about projective planes in "week106". The basic idea is
to take the ordinary plane and add some points at infinity so that every
pair of distinct lines intersects in exactly one point. Lines that were
parallel in the ordinary plane will intersect at one of the points at
infinity. This simplifies the axioms of projective geometry.
But what exactly do I mean by "the ordinary plane"? Well, ever since
Descartes, most people think of the plane as R^2, which consists of
ordered pairs of real numbers. But algebraists also like to use C^2,
consisting of ordered pairs of complex numbers. For that matter, you
could take any field F  like the rational numbers, or the integers
modulo a prime  and use F^2. Algebraic geometers call this sort of
thing an "affine plane".
A projective plane is a bit bigger than an affine plane. For this,
start with the 3dimensional vector space F^3. Then define the
projective plane over F, denoted FP^2, to be the space of lines through
the origin in F^3. You can show the projective plane is the same as
the affine plane together with extra points, which play the role of
"points at infinity".
In fact, you can generalize this a bit  you can make sense of the
projective plane over F whenever F is a division ring! A division ring
is a like a field, but where multiplication isn't necessarily
commutative. The best example is the quaternions. In "week106"
I talked about the real, complex and quaternionic projective planes,
their symmetry groups, and their relation to quantum mechanics. Here's
a good book about this stuff, emphasizing the physics applications:
1) V. S. Varadarajan, Geometry of Quantum Mechanics, SpringerVerlag,
Berlin, 2nd ed., 1985.
So far, so good. But there's another approach to projective planes
that's even more general. This approach goes back to Euclidean
geometry: it's based on a list of axioms. In this approach, a
projective plane consists of a set of "points", a set of "lines", and
a relation which tells us whether or not a given point "lies on" a
given line. I'm putting quotes around all these words, because in this
approach they are undefined terms. All we get to work with are the
following axioms:
A) Given two distinct points, there exists a unique line that both
points lie on.
B) Given two distinct lines, there exists a unique point that lies on
both lines.
C) There exist four points, no three of which lie on the same line.
D) There exist four lines, no three of which have the same point lying
on them.
Actually we can leave out either axiom C) or axiom D)  the rest of the
axioms will imply the one we leave out. It's a nice little exercise to
convince yourself of this. I put in both axioms just to make it obvious
that this definition of projective plane is "selfdual". In other
words, if we switch the words "point" and "line" and switch who lies on
who, the definition stays the same!
Duality is one of the great charms of the theory of projective planes:
whenever you prove any theorem, you get another one free of charge with
the roles of points and lines switched, thanks to duality. There are
lots of different kinds of "duality" in mathematics, but this is
probably the granddaddy of them all.
Now, it's easy to prove that starting from any division ring F, we get a
projective plane FP^2 satisfying the above axioms. The fun part is to
try to go the other way! Starting from a structure satisfying the
above axioms, can you cook up a division ring that it comes from?
Well, starting from a projective plane, you can try to recover a
division ring as follows. Pick a line and throw out one point  and
call that point "the point at infinity". What's left is an "affine
line"  let's call it L. Let's try to make L into a division ring. To
do this, we first need to pick two different points in L, which we call
0 and 1. Then we need to cook up rules for adding and multiplying
points on L.
For this, we use some tricks invented by the ancient Greeks!
This should not be surprising. After all, those dudes thought about
arithmetic in very geometrical ways. How can you add points on a line
using the geometry of the plane? Just ask any ancient Greek, and here's
what they'll say:
First pick a line L' that's parallel to L  meaning that L and L'
intersect only at the point at infinity. Then pick a line M that
intersects L at the point 0 and L' at some point which we call 0'.
We get a picture like this:

0' L'


M

0 L

Then, to add two points x and y on L, draw this picture:
 
0' L'
 \  \
 \  \
M \N M' \N'
 \  \
0xyz L
 \  \
In other words, draw a line M' parallel to M through the point y,
draw a line N through x and 0', and draw a line N' parallel to N and
going through the point where M' and L' intersect. L and N' intersect
at the point called z... and we define this point to be x + y!
This is obviously the right thing, because the two triangles in the
picture are congruent.
What about multiplication? Well, first draw a line L' that intersects
our line L only at the point 0. Then draw a line M from the point 1 to
some point 1' that's on L' but not on L:
L'/
/
/
1'
/
/ 
/ 
01 L

M

Then, to multiply x and y, draw this picture:
/
/\
/  \
/  \
L'/  \
/  \
/  \
1'  \
/\  \
/  \N  \N'
/  \  \
01xyz L
 \  \
M M'
 
In other words, draw a line N though 1' and x, draw a line M' parallel
to M through the point y, and draw a line N' parallel to N through the
point where L' and M' intersect. L and N' intersect at the point called
z... and we define this point to be xy!
This is obviously the right thing, because the triangle containing
the points 1 and x is similar to the triangle containing y and z.
So, now that we've cleverly figured out how to define addition and
multiplication starting from a projective plane, we can ask: do we
get a division ring?
And the answer is: not necessarily. It's only true if our projective
plane is "Desarguesian". This is a special property named after an
old theorem about the real projective plane, proved by Desargues.
A projective plane is Desarguesian if Desargues' theorem holds for
this plane.
But wait  there's an even more basic question we forgot to ask!
Namely: was our ancient Greek method of defining addition and
multiplication independent of the choices we made? We needed to pick
some points and lines to get things going. If you think about it hard,
these choices boil down to picking four points, no three of which lie on
a line  exactly what axiom C) guarantees we can do.
Alas, it turns out that in general our recipe for addition and
multiplication really depends on *how* we chose these four points.
But if our projective plane is Desarguesian, it does not!
In fact, if we stick to Desarguesian projective planes, everything works
very smoothly. For any division ring F the projective plane FP^2 is
Desarguesian. Conversely, starting with a Desarguesian projective
plane, we can use the ancient Greek method to cook up a division ring F.
Best of all, these two constructions are inverse to each other  at
least up to isomorphism.
At this point you should be pounding your desk and yelling "Great  but
what does `Desarguesian' mean? I want the nittygritty details!"
Okay....
Given a projective plane, define a "triangle" to be three points that
don't lie on the same line. Now suppose you have two triangles xyz and
x'y'z'. The sides of each triangle determine three lines, say LMN and
L'M'N'. If we're really lucky, the line through x and x', the line
through y and y', and the line through z and z' will all intersect at
the same point. We say that our projective plane is "Desarguesian" if
whenever this happens, something else happens: the intersection of L
and L', the intersection of M and M', and the intersection of N and N'
all lie on the same line.
If you have trouble visualizing what I just said, take a look at this
webpage, which also gives a proof of Desargues' theorem for the real
projective plane:
1) Roger Mohr and Bill Trigs, Desargues' Theorem,
http://spigot.anu.edu.au/people/samer/Research/Doc/ECV_Tut_Proj_Geom/node25.html
Desargues' theorem is a bit complicated, but one cool thing is that
its converse is its dual. This is easy to see if you stare at it:
"three lines intersecting at the same point" is dual to "three points
lying on the the same line". Even cooler, Desargues' theorem implies
its own converse! Thus the property of being Desarguesian is selfdual.
Another nice fact about Desarguesian planes concerns collineations. A
"collineation" is a map from a projective plane to itself that preserves
all lines. Collineations form a group, and this group acts on the set
of all "quadrangles"  a quadrangle being a list of four points, no
three of which lie on a line. Axiom C) says that every projective
plane has at least one quadrangle. It turns out that if a projective
plane is Desarguesian, the group of collineations acts transitively on
the set of quadrangles: given any two quadrangles, there's a collineation
carrying one to the other. This is the reason why the ancient Greek trick
for adding and multiplying doesn't depend on the choice of quadrangle when
our projective plane is Desarguesian!
An even more beautiful fact about Desarguesian planes concerns their
relation to higher dimensions. Just as we defined projective planes
through a list of axioms, we can also define projective spaces of any
dimension n = 1,2,3,.... The simplest example is FP^n  the space of
lines through the origin in F^{n+1}, where F is some division ring. The
neat part is that when n > 2, this is the *only* example. Moreover, any
projective plane sittting inside one of these higherdimensional projective
spaces is automatically Desarguesian! So the nonDesarguesian projective
planes are really freaks of dimension 2.
All this is very nice. But there are some obvious further questions,
namely: what's special about projective planes that actually come from
*fields*, and what can we say about nonDesarguesian projective planes?
The key to the first question is an old theorem proved by the last of
the great Greek geometers, Pappus, in the 3rd century CE. It turns out
that in any projective plane coming from a field, the Pappus Theorem
holds. Conversely, any projective plane satisfying the Pappus Theorem
comes from a unique field. We call such projective planes "Pappian".
The Pappus theorem will be too scary if I explain it using only words,
so I'll tell you to look at a picture instead. The fun thing about this
picture is that you can move the red and green points around with your
mouse and see how things change:
2) Pappus' theorem (a JavaSketchPad demo by MathsNet),
http://www.anglia.co.uk/education/mathsnet/dynamic/pappus.html
Now, what about the nonDesarguesian projective planes? If we try to
get a division ring from an *arbitrary* projective plane, we fail
miserably. However, we can still define addition and multiplication
using the tricks described above. These operations depend crucially our
choice of a quadrangle. But if we list all the axioms these operations
satisfy, we get the definition of an algebraic gadget called a "ternary
ring".
They're called "ternary rings" because they're usually described in
terms of a ternary operation that generalizes xy + z. But the precise
definition is too depressing for me give here. It's a classic example
of what James Dolan calls "centipede mathematics", where you take a
mathematical concept and see how many legs you can pull off before it
can no longer walk. A ternary ring is like a division ring that can
just barely limp along on its last legs.
I'm not a big fan of centipede mathematics, but there is one really
nice example of a ternary ring that isn't a division ring. Namely,
the octonions! These are almost a division ring, but their multiplication
isn't associative.
I already talked about the octonions in "week59", "week61", "week104"
and "week105". In "week106", I explained how you can define OP^2, the
projective plane over the octonions. This is the best example of a
nonDesarguesian projective plane. One reason it's so great is that
that its group of collineations is E6. E6 is one of the five
"exceptional simple Lie groups"  mysterious and exciting things that
deserve all the study they can get!
Next I want to talk about the relation between projective geometry
and the *other* exceptional Lie groups, but first let me give you
some references. To start, here's a great book on projective planes
and all the curious centipede mathematics they inspire:
3) Frederick W. Stevenson, Projective Planes, W. H. Freeman and Company,
San Francisco, 1972.
You'll learn all about nearfields, quasifields, Moufang loops, Cartesian
groups, and so on.
Much of the same material is covered in these lectures by Hall,
which are unfortunately a bit hard to find:
4) Marshall Hall, Projective Planes and Other Topics, California
Institute of Technology, Pasadena, 1954.
For a more distilled introduction to the same stuff,
try the last chapter of Hall's book on group theory:
5) Marshall Hall, The Theory of Groups, Macmillan, New York, 1959.
If you're only interested in Desarguesian projective planes, try
this:
6) Robin Hartshorne, Foundations of Projective Geometry, Benjamin,
New York, 1967.
In particular, this book gives a nice account of the collineation group
in the Desarguesian case. The punchline is simple to state, so I'll
tell you. Suppose F is a division ring. Then the collineation
group of FP^2 is generated by two obvious subgroups: PGL(3,F) and
the automorphism group of F. The intersection of these two subgroups
is the group of inner automorphisms of F.
If the above references are too intense, try this leisurely, literate
introduction to the subject first:
7) Daniel Pedoe, An Introduction to Projective Geometry, Macmillan,
New York, 1963.
And you're really interested in the *finite* projective planes, you
can try this reference, which assumes very little knowledge of algebra:
8) A. Adrian Albert and Reuben Sandler, An Introduction to Finite
Projective Planes, Holt, Rinehart and Winston, New York, 1968.
For a nice online introduction to projective geometry over the real
numbers and its applications to image analysis, try this:
9) Roger Mohr and Bill Triggs, Projective geometry for image analysis,
http://spigot.anu.edu.au/people/samer/Research/Doc/ECV_Tut_Proj_Geom/node1.html
Finally, for interesting relations between projective geometry
and exceptional Lie groups, try this:
10) J. M. Landsberg and L. Manivel: The projective geometry of
Freudenthal's magic square, preprint available as math.AG/9908039.
The FreudenthalTits magic square is a strange way of describing
most of the exceptional Lie groups in terms of the real numbers,
complex numbers, quaternions and octonions. In the usual way of
describing it, you start with two of these division algebras, say
F and F'. Then let J(F) be the space of 3x3 selfadjoint matrices
with coefficients in F. This is a Jordan algebra with the product
xy + yx. As mentioned in "week106", Jordan algebras have a lot to
do with projective planes. In particular, the nontrivial projections
in J(F) correspond to the 1 and 2dimensional subspaces of F^3,
and thus to the points and lines in the projective plane PF^2.
Next, let J_0(F) be the subspace of J(F) consisting of the *traceless*
selfadjoint matrices. Also, let Im(F') be the space of pure imaginary
element of K'. Finally, let the magic Lie algebra M(F,F') be given by
M(K,K') = Der(J(K)) + J_0(K) x Im(K') + Der(K')
Here + stands for direct sum, x stands for tensor product, and Der
stands for the space of derivations of the algebra in question.
It's actually sort of tricky to describe how to make M(F,F') into
a Lie algebra, and I'm sort of tired, so I'll wimp out and tell
you to read this stuff:
11) Hans Freudenthal, Lie groups in the foundations of geometry,
Adv. Math. 1 (1964) 143.
12) Jacques Tits, Algebres alternatives, algebres de Jordan et
algebres de Lie exceptionelles, Proc. Colloq. Utrecht, vol. 135, 1962.
13) R. D. Schafer, Introduction to Nonassociative Algebras, Academic
Press, 1966.
By the way, the paper by Freudenthal is a really mindbending mix of
Lie theory and axiomatic projective geometry, definitely worth looking
at. Anyway, if you do things right you get the following square of Lie
algebras M(F,F'):
F = R F = C F = H F = O
F' = R A1 A2 C3 F4
F' = C A2 A2+A2 A5 E6
F' = H C3 A5 B6 E7
F' = O F4 E6 E7 E8
Here R, C, H and O stand for the reals, complexes, quaternions and
octonions. If you don't know what all the Lie algebras in the square
are, check out "week64". (I should admit that the above square is
not very precise, because I don't say which real forms of the Lie
algebras in question are showing up.)
The first fun thing about this square is that F4, E6, E7 and E8 are
four of the five exceptional simple Lie algebras  and the fifth
one, G2, is just Der(O). So all the exceptional Lie algebras are
related to the octonions! And the second fun thing about this square
is that it's symmetrical along the diagonal, even though this is
not at all obvious from the definition. This is what makes the square
truly "magic".
I don't really understand the magic square, but it's on my todo
list. That's one reason I'm glad there's a new paper out that
describes the magic square in a way that makes its symmetry manifest:
14) C. H. Barton and A. Sudbery, Magic squares of Lie algebras,
preprint available as math.RA/0001083.
It also generalizes the magic square in a number of directions. But
what I really want is for the connection between projective planes,
division algebras, exceptional Lie groups and the magic square to
becomes truly *obvious* to me. I'm a long way from that point.
Quote of the week:
The reader should not attempt to form a mental picture of a closed
straight line.  Frank Ayres, Jr., Projective Geometry

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