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.Among other things, this book has wonderful charts showing the development of each numeral. You can see, for example, how the primitive numeral
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.
____ ____ ____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. But of course, there's no reason to believe it is.
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 h3(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 V8, the right-handed spinor representation S8+, and the left-handed spinor representation S8-.
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 h3(O) and identify them with elements of the three 8-dimensional irreps of Spin(8), we get
h3(O) = R3 + V8 + S8+ + S8-A little calculation then reveals a wonderful fact: while superficially the Jordan product in h3(O) is built using the structure of O as a normed division algebra, it can actually be defined using just the natural map
t: V8 x S8+ x S8- → Rand the inner products on these 3 spaces. It follows that any element of Spin(8) gives an automorphism of h3(O). Indeed, Spin(8) becomes a subgroup of Aut(h3(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:
S9 = S8+ + S8- V9 = R + V8This gives an isomorphism
h3(O) = R2 + V9 + S9and in fact the product in h3(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(h3(O)).
This does not exhaust all the symmetries of h3(O), since there are other automorphisms coming from the permutation group on 3 letters, which acts on (a,b,c) in R3 and (x,y,z) in O3 in an obvious way. Also, any matrix g in the orthogonal group O(3) acts by conjugation as an automorphism of h3(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 h3(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 h3(O). To see this, first note that the rotation group, and thus Spin(9), acts transitively on the unit sphere in the vector representation V9. This means we can use an automorphism in our Spin(9) subgroup to bring any element of h3(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 V9 is isomorphic to Spin(8). When we restrict the representation S9 to this subgroup it splits as S8+ + S8-, and with some work one can show that Spin(8) acts on S8+ + S8- = O2 in such a way that any element (y,z) in O2 can be carried to an element with both components real. The final step is to take our element of h3(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 h3(O). Here is where our ability to diagonalize matrices in h3(O) via automorphisms comes in handy. Up to automorphism, every projection in h3(O) looks like one of these four guys:
( 0 0 0 ) ( 0 0 0 ) = p0 ( 0 0 0 ) ( 1 0 0 ) ( 0 0 0 ) = p1 ( 0 0 0 ) ( 1 0 0 ) ( 0 1 0 ) = p2 ( 0 0 0 ) ( 1 0 0 ) ( 0 1 0 ) = p3 ( 0 0 1 )Now, the trace of a matrix in h3(O) is invariant under automorphisms, because we can define it using only the Jordan algebra structure:
tr(a) = (1/3) tr(La)where La is left multiplication by a. It follows that the trace of any projection in h3(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
p0 < p1 < ... < pd = pIn 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 h3(O) the dimension equals the trace.
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 p0 < p1 < p2 < p3, 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 h3(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 = |ψ> <ψ| = ( xx* xy* xz* ) ( yx* yy* yz* ) ( zx* zy* zz* )where
|ψ> = (x,y,z)is a unit vector in O3 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 OP2 to stand for the set of points in the octonionic projective plane. Given any nonzero element (x,y,z) in O3 with (xy)z = x(yz), we can normalize it and then use the above formula to obtain a point of OP2, which we call [(x,y,z)]. We can make OP2 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 OP2.
Clearly the manifold OP2 is 16-dimensional. The lines in OP2 are copies of OP1, 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 OP1, 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 OP1 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!