Also available at http://math.ucr.edu/home/baez/week163.html
December 31, 2000
This Week's Finds in Mathematical Physics (Week 163)
John Baez
If you think numbers start with the number 1, you probably think the
millennium is ending now. I think it ended last year... but either
way, now is a good time to read this book:
1) Georges Ifrah, The Universal History of Numbers from Prehistory to
the Invention of the Computer, Wiley, New York, 2000.
On the invention of zero:
Most peoples throughout history failed to discover the rule of
position, which was discovered in fact only four times in the
history of the world. (The rule of position is the principle in
which a 9, let's say, has a different magnitude depending on
whether it comes in first, second, third... position in a numerical
expression.) The first discovery of this essential tool of
mathematics was made in Babylon in the second millennium BCE.
It was then rediscovered by the Chinese arithmeticians at around
the start of the Common Era. In the third to fifth centuries CE,
Mayan astronomers reinvented it, and in the fifth century CE it
was rediscovered for the last time, in India.
Obviously, no civilization outside of these four ever felt the
need to invent zero; but as soon as the rule of position became
the basis for a numbering system, a zero was needed. All the same,
only three of the four (the Babylonians, the Mayans, and the
Indians) managed to develop this final abstraction of number;
the Chinese only acquired it through Indian influences. However,
the Babylonian and Mayan zeroes were not conceived of as numbers,
and only the Indian zero had roughly the same potential as the
one we use nowadays. That is because it is indeed the Indian zero,
transmitted to us through the Arabs together with the number-symbols
that we call Arabic numerals and which are in reality Indian numerals,
with their appearance altered somwhat by time, use and travel.
Among other things, this book has wonderful charts showing the
development of each numeral. You can see, for example, how the
primitive numeral
____
____
____
slowly evolved to our modern "3". Hmm - how come this doesn't feel
like progress?
Now, I usually keep my eyes firmly focused on the beauties of nature,
but once in a millennium I feel the need to engage in some politics.
So....
In "week155" I talked a lot about polyhedra and their 4-dimensional
generalizations, and I referred to Eric Weisstein's online math
encyclopedia since it had lots of nice pictures. Now this website
has been closed down, thanks to a lawsuit by the people at CRC Press:
2) Frequently asked questions about the MathWorld case,
http://mathworld.wolfram.com/docs/faq.html
Weisstein published a print version of his encyclopedia with CRC press,
but now they claim to own the rights to the online version as well.
So I urge you all to remember this: when dealing with publishers, never
sign away the electronic rights on your work unless you're willing to
accept the consequences!
For example, suppose you write a math or physics paper and put it on the
preprint archive, and then publish it in a journal. They'll probably
send you a little form to sign where you hand over the rights to this
work - including the electronic rights. If you're like most people,
you'll sign this form without reading it. This means that if they feel
like it, they can now sue you to make you take your paper off the
preprint archive! Journals don't do this yet, but as they continue
becoming obsolete and keep fighting ever more desperately for their
lives, there's no telling what they'll do. Corporations everywhere are
taking an increasingly aggressive line on intellectual property rights -
as the case of Weisstein shows.
So what can you do? Simple: don't agree to it. When you get this form,
cross out any sentences you refuse to agree to, put your initials by
these deletions, and sign the thing - indicating that you agree to the
*other* stuff! Keep a copy. If they complain, ask them how much these
electronic rights are worth.
Basically, I think it's time for academics to take more responsibility
about keeping their work easily accessible.
There are lots of things you can do. One of the easiest is to stop
refereeing for ridiculously expensive journals. Journal prices bear
little relation to the quality of service they provide. For example,
the Elsevier-published journal "Nuclear Physics B" costs $12,596 per
year for libraries, or $6,000 for a personal subscription. The
comparable journal "Advances in Theoretical and Mathematical Physics"
costs $300 for libraries or $80 for a personal subscription - and
access to the electronic version is free. So when Nuclear Physics B
asks me to referee manuscripts, I now say "Sorry, I'll wait until your
prices go down."
In fact, I no longer referee articles for any journals published by
Elsevier, Kluwer, or Gordon & Breach. If you've looked at their prices,
you'll know why. G&B has even taken legal action against the American
Institute of Physics, the American Physical Society, and the American
Mathematical Society for publishing information about journal prices!
3) Gordon and Breach et al v. AIP and APS, brief of amici curiae of the
American Library Association, Association of Research Libraries and
the Special Library Association, http://www.arl.org/scomm/gb/amici.html
4) AIP/APS prevail in suit by Gordon and Breach, G&B to appeal,
http://www.arl.org/newsltr/194/gb.html
Of course, the ultimate solution is to support the math and physics
preprint archives, and figure out ways to decouple the refereeing
process from the distribution process.
Okay, enough politics. I was thinking about 4-dimensional polytopes,
and Eric Weisstein's now-defunct website... but what got me going in the
first place was this:
5) John Stilwell, The story of the 120-cell, AMS Notices 48 (January 2001),
17-24.
The 120-cell is a marvelous 4-dimensional shape with 120 regular
dodecahedra as faces. I talked about it in "week155", but this article
is full of additional interesting information. For example, Henri
Poincare once conjectured that every compact 3-manifold with the same
homology groups as a 3-sphere must *be* a 3-sphere. He later proved
himself wrong by finding a counterexample: the "Poincare homology
3-sphere". This is obtained by identifying the opposite faces of the
dodecahedron in the simplest possible way. What I hadn't known is that
the fundamental group of this space is the "binary icosahedral group", I.
This is the 120-element subgroup of SU(2) consisting of all elements
that map to rotational symmetries of the icosahedron under the
two-to-one map from SU(2) to SO(3). Now SU(2) is none other than the
3-sphere... so it follows that SU(2)/I is the Poincare homology 3-sphere!
When cosmologists study the possility that universe is finite in size,
they usually assume that space is a 3-sphere. In this scenario,
barring sneaky tricks, it's likely that the universe would recollapse
before light could get all the way around the universe. But there's no
strong reason to favor this topology. Some people have checked to see
whether space is a 3-dimensional torus. In such a universe, light might
wrap all the way around - so you might see the same bright quasars by
looking in various different directions! People have looked for this
effect but not seen it. This doesn't rule out a torus-shaped universe,
but it puts a limit on how small it could be.
In fact, some physicists have even considered the possibility that space
is a Poincare homology 3-sphere! Can light go all the way around in
this case? I don't know. If so, we might see bright quasars in a
pretty dodecahedral pattern.
Amusingly, Plato hinted at something resembling this in his "Timaeus":
6) Plato, Timaeus, translated by B. Jowett, in The Collected Dialogues,
Princeton U. Press, Princeton, 1969 (see line 55c).
This dialog is one the first attempts at doing mathematical physics.
In it, the Socrates character guesses that the four elements earth,
air, water and fire are made of atoms shaped like four of the five
Platonic solids: cubes, octahedra, icosahedra and tetrahedra, respectively.
Why? Well, fire obviously feels hot because of those pointy little
tetrahedra poking you! Water is liquid because of those round little
icosahedra rolling around. Earth is solid because of those little cubes
packing together so neatly. And air... well, ahem... we'll get back to
you on that one.
BUT WHAT ABOUT THE DODECAHEDRON? On this topic, Plato makes only the
following cryptic remark: "There was yet a fifth combination which God
used in the delineation of the universe with figures of animals."
Huh??? I think this is a feeble attempt to connect the 12 sides of the
dodecahedron to the 12 signs of the zodiac. After all, lots of the
signs of the zodiac are animals. The word "zodiac" comes from the
Greek phrase "zodiakos kuklos", or "circle of carved figures" - where
"zodiakos" or "carved figure" is really the diminutive of "zoion",
meaning "animal". There may even be a connection between the
dodecahedron and the "quintessence": the fifth element, of which the
heavenly bodies were supposedly made. I know, this is all pretty weird,
but there seems to be some tantalizingly murky connection between the
dodecahedron and the heavens in Greek cosmology.... so it would be cool
if space turned out to be a Poincare homology 3-sphere.
Okay, enough goofing around. Now let me talk a bit about the
exceptional Jordan algebra and the octonionic projective plane. I'll
basically pick up where I left off in "week162" - but you might want to
reread "week61", "week106" and "week145" to prepare yourself for the
weirdness to come. Also, keep in mind the following three facts about
the number 3, which fit together in a spooky sort of synergy that makes
all the magic happen:
i) An element of h_3(O) is a 3x3 hermitian matrix with octonionic
entries, and thus consists of 3 octonions and 3 real numbers:
( a z* y* )
( z b x ) a,b,c in R, x,y,z in O
( y x* c )
ii) The octonions arise naturally from "triality": the relation between
the three 8-dimensional irreps of Spin(8), i.e. the vector representation
V(8), the right-handed spinor representation S(8,+), and the left-handed
spinor representation S(8,-).
iii) The associative law (xy)z = x(yz) involves 3 variables.
Let's see how it goes.
First, if we take the 3 octonions in our element of h_3(O) and identify
them with elements of the three 8-dimensional irreps of Spin(8), we get
h_3(O) = R^3 + V(8) + S(8,+) + S(8,-)
A little calculation then reveals a wonderful fact: while superficially
the Jordan product in h_3(O) is built using the structure of O as a normed
division algebra, it can actually be defined using just the natural map
t: V(8) x S(8,+) x S(8,-) ---> R
and the inner products on these 3 spaces. It follows that any element
of Spin(8) gives an automorphism of h_3(O). Indeed, Spin(8) becomes a
subgroup of Aut(h_3(O)).
So the exceptional Jordan algebra has a lot to do with geometry in 8
dimensions - that's not surprising. What's surprising is that it also
has a lot to do with geometry in 9 dimensions! When we restrict the
spinor and vector representations of Spin(9) to the subgroup Spin(8),
they split as follows:
S(9) = S(8,+) + S(8,-)
V(9) = R + V(8)
This gives an isomorphism
h_3(O) = R^2 + V(9) + S(9)
and in fact the product in h_3(O) can be described in terms of natural
maps involving scalars, vectors and spinors in 9 dimensions. It follows
that Spin(9) is also a subgroup of Aut(h_3(O)).
This does not exhaust all the symmetries of h_3(O), since there are
other automorphisms coming from the permutation group on 3 letters,
which acts on (a,b,c) in R^3 and (x,y,z) in O^3 in an obvious way.
Also, any matrix g in the orthogonal group O(3) acts by conjugation as
an automorphism of h_3(O); since the entries of g are real, there is no
problem with nonassociativity here. The group Spin(9) is 36-dimensional,
but the full automorphism group h_3(O) is 52-dimensional. In fact, it
is the exceptional Lie group F4!
However, we can already do something interesting with the automorphisms
we have: we can use them to diagonalize any element of h_3(O). To
see this, first note that the rotation group, and thus Spin(9), acts
transitively on the unit sphere in the vector representation V(9).
This means we can use an automorphism in our Spin(9) subgroup to bring
any element of h_3(O) to the form
( a z* y* )
( z b x )
( y x* c )
where x is *real*. The next step is to apply an automorphism that
makes y and z real while leaving x alone. To do this, note that the
subgroup of Spin(9) fixing any nonzero vector in V(9) is isomorphic to
Spin(8). When we restrict the representation S(9) to this subgroup
it splits as S(8,+) + S(8,-), and with some work one can show that
Spin(8) acts on S(8,+) + S(8,-) = O^2 in such a way that any element
(y,z) in O^2 can be carried to an element with both components real.
The final step is to take our element of h_3(O) with all real entries
and use an automorphism to diagonalize it. We can do this by
conjugating it with a suitable matrix in O(3).
To understand the octonionic projective plane, we need to understand
projections in h_3(O). Here is where our ability to diagonalize
matrices in h_3(O) via automorphisms comes in handy. Up to automorphism,
every projection in h_3(O) looks like one of these four guys:
p_0 = ( 0 0 0 )
( 0 0 0 )
( 0 0 0 )
p_1 = ( 1 0 0 )
( 0 0 0 )
( 0 0 0 )
p_2 = ( 1 0 0 )
( 0 1 0 )
( 0 0 0 )
p_3 = ( 1 0 0 )
( 0 1 0 )
( 0 0 1 )
Now, the trace of a matrix in h_3(O) is invariant under automorphisms,
because we can define it using only the Jordan algebra structure:
tr(a) = (1/3) tr(L_a)
where L_a is left multiplication by a. It follows that the trace
of any projection in h_3(O) is 0, 1, 2, or 3.
Remember from "week162" that the "dimension" of a projection p in a
formally real Jordan algebra is the largest number d such that there's
a chain of projections
p_0 < p_1 < ... < p_d = p
In favorable cases, like the exceptional Jordan algebra, the dimension-1
projections become the points of a projective plane, while the dimension-2
projections become the lines. But what's a practical way to compute
the dimension of a projection? Well, in h_3(O) the dimension equals the
trace.
Why?
Well, clearly the dimension is less than or equal to the trace, since
p < q implies tr(p) < tr(q), and the trace goes up by integer steps.
But on the other hand, the trace is less than or equal to the dimension.
To see this it suffices to consider the four projections shown above,
since both trace and dimension are invariant under automorphisms. Since
p_0 < p_1 < p_2 < p_3, it is clear that for these projections the trace
is indeed less than or equal to the rank.
So: the points of the octonionic projective plane are the projections
with trace 1 in h_3(O), while the lines are projections with trace 2.
A brutal calculation in Reese Harvey's book:
7) F. Reese Harvey, Spinors and Calibrations, Academic Press, Boston, 1990.
reveals that any projection with trace 1 has the form
p = |psi> = (x,y,z)
is a unit vector in O^3 for which (xy)z = x(yz). This is supposed to
remind you of stuff about spinors and the heavenly sphere in "week162".
On the other hand, any projection with trace 2 is of the form 1 - p
where p has trace 1. This sets up a one-to-one correspondence between
points and lines in the octonionic projective plane. If we use this
correspondence to think of both as trace-1 projections, the point p
lies on the line q if and only if p < 1 - q. Of course, p < 1 - q
iff q < 1 - p. The symmetry of this relation means the octonionic
projective plane is self-dual! This is also true of the real, complex
and quaternionic projective planes. In all cases, the operation that
switches points and lines corresponds in quantum logic to "negation".
Let's use OP^2 to stand for the set of points in the octonionic
projective plane. Given any nonzero element (x,y,z) in O^3 with (xy)z
= x(yz), we can normalize it and then use the above formula to obtain a
point of OP^2, which we call [(x,y,z)]. We can make OP^2 into a smooth
manifold by covering it with three coordinate charts: one containing
all points of the form [(x,y,1)], one containing all points of the form
[(x,1,z)], and one containing all points of the form [(1,y,z)].
Checking that this works is a simple calculation. The only interesting
part is to make sure that whenever the associative law might appear
necessary, we can either use the weaker equations
(xx)y = x(xy)
(xy)x = x(yx)
(yx)x = y(xx)
which still hold for the octonions, or else the fact that only triples
with (xy)z = x(yz) give points [(x,y,z)] in OP^2.
Clearly the manifold OP^2 is 16-dimensional. The lines in OP^2 are copies
of OP^1, and thus 8-spheres. It is also good to work out the space
of lines going through any point. Here we can use self-duality: since
the space of all points lying on any given line is a copy of OP^1, so
is the space of all lines on which a given point lies! So the space
of lines through a point is also an 8-sphere. Everything is very pretty.
If we give OP^1 the nicest possible metric, its isometry group is F4:
just the automorphism group of the exceptional Jordan algebra. However,
the group of "collineations" - i.e., line-preserving transformations -
is a form of the 78-dimensional exceptional Lie group E6. From stuff
explained last week, the subgroup of collineations that map a point
p to itself and also map the line 1 - p to itself is isomorphic to
Spin(9,1). This gives a nice embedding of Spin(9,1) in this form of E6.
So the octonionic projective plane is also related to 10-dimensional
*spacetime* geometry.
I hope I've got that last part right.... ultimately, this is supposed to
explain why various different theories of physics formulated in 10d
spacetime wind up being related to the exceptional Lie groups! But I'm
afraid that so far, I'm just struggling to understand the basic geometry.
Happy New Year!
----------------------------------------------------------------------
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mathematics and physics, as well as some of my research papers, can be
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