October 29, 2001

This Week's Finds in Mathematical Physics (Week 172)

John Baez

I recently went to a conference on "Discrete Random Geometries and Quantum Gravity", organized by Renate Loll:

1) Discrete Random Geometries and Quantum Gravity, http://www1.phys.uu.nl/Symposion/EUWorkshop.htm

She was one of the people who first gave me the courage to work on quantum gravity. I'd been interested in it for a long time, but I didn't like how string theory relied on supersymmetry and a background metric, so I didn't know any approach that looked promising until I saw her give a talk on loop quantum gravity at a conference in Seattle in the early 1990s. She was interested in numerical simulation of quantum gravity models even back then, and by now she's one of the top experts on this subject. But it's extremely hard to get permanent positions in quantum gravity, especially in Europe, so I was happy when she recently got a job at the University of Utrecht. To kick off her stay there, she threw this conference!

I like to read "Wired" magazine when I'm on long airplane trips. On my flight to Amsterdam, I found this interesting article:

2) Wil McCarthy, Ultimate alchemy, Wired, October 2001, 150.

It's about people are using "quantum dots" to make "artificial atoms". A quantum dot is a tiny speck of conductive material that can be used as a potential well holding one or more electrons in a bound state. Such bound states are a lot like atoms! However, the ones people have made so far are about 50 times bigger than actual atoms, because they are more loosely bound. This also means that they ionize more easily, so they need to be kept very cold.

However, they can have more electrons than normal atoms, since they aren't limited by the tendency of large nuclei to undergo radioactive decay, or ultimately, somewhere around element 137, the tendency of strong electric fields to "spark the vacuum" by creation of particle-antiparticle pairs - a quantum field theory effect that's not included in the bare-bones Schroedinger equation. So, someday we may learn how the periodic table goes up to, say, element 500! I've sometimes imagined decadent future chemists studying such elements on the computer, just for the fun of it... but now perhaps they'll do it with "artificial atoms".

Now, McCarthy is a science fiction writer, so he imagines more dramatic applications of quantum dots, like "programmable matter" - a gadget whose surface can, say, turn from lead to gold at the flick of a switch. Personally I don't see how to get these tricks to work at room temperature until we make artificial atoms almost as small as real ones, which I don't see how to do without them being... atoms! But even so, I believe there will be some cool technological applications of quantum dots.

For more on quantum dots by experts on the subject, try these papers:

3) Marc Kastner, Artificial atoms, Physics Today 46 (1993), 24. Also available at http://web.mit.edu/physics/people/marc_kastner.htm

4) Leo Kouwenhoven and Charles Marcus, Quantum dots, Physics World, June 1998. Also available at http://marcuslab.harvard.edu/

Unfortunately I didn't have access to these papers on my flight from Los Angeles to Amsterdam. It takes 10 hours, so I had to read a lot more to keep from going insane with boredom. Even the latest news about bioterrorism and bombings was not enough to keep me entertained. (By the way, I predict that a highly contagious virus will sweep the United States and kill about 20,000 people within the next few months. It's called "influenza", and that's the average number of Americans who die from it each year. I plan to call the FBI and warn them about this.)

So, I had to hit the serious mathematical physics:

5) Terry Gannon, Monstrous moonshine and the classification of CFT, in Conformal Field Theory: New Non-Perturbative Methods in String and Field Theory, Yavuz Nutku, Cihan Saclioglu and Teoman Turgut, eds., Perseus Publishing, 2000.

This is a very pleasant 66-page review article on "monstrous moonshine", which is what people call the relation between the Monster group and modular forms. Someday I'll have to say a lot more about this; for now see "week66" if you have no idea what I'm talking about. Gannon's article is full of juicy mathematical tidbits and pieces of wisdom. He even gives a new explanation of why the number 24 is so important throughout mathematics and string theory. If x² = 1 mod n, then x must be relatively prime to n... and 24 is the largest integer for which the converse holds! Alas, Gannon does not explain how this relates to the other magic properties of this number, some of which are listed in "week124". Does anyone see the connection?

At the conference, one of my favorite talks was by Sergeui Dorogovtsev, on "Geometry of Evolving Random Networks". A directed graph is a bunch of nodes connected by edges with little arrows on them. A nice example is the world-wide web, where the nodes are webpages and the edges are links. Various people have noticed that in naturally evolving directed graphs, the number of edges to or from a given node is distributed roughly according to a power law. For example, on the World-Wide Web, the number of sites having n links to them is roughly proportional to

n^-2.1

while the number of sites having n links coming from them is roughly proportional to

n^-2.7

This differs from the simple models of random graphs most studied by mathematicians, for which these quantities often follow a Poisson distribution. But recently people have been coming up with new models of evolving graphs that have this power-law behavior. The trick is to take into account the fact that "popularity is attractive". The simplest model uses undirected graphs: keep adding new nodes one at a time, and let the probability that your new node has an edge to any existing node be proportional to the number of edges already attached to the existing node. Following this rule, you'll build up a big random graph with the power law behavior

n^-3.

For more details see this fascinating paper:

6) Sergeui N. Dorogovtsev and J.F.F. Mendes, Evolving networks, available at cond-mat/0106144.

I really love the chart on page 11! It shows the general structure of a typical naturally arising large directed graph such as the World-Wide Web. The picture is worth a thousand words, but let me try to explain it:

First, a large fraction of the nodes lie in the "giant strongly connected component", or GSCC. This is the biggest set of nodes where you can get between any two by following a sequence of edges and going forwards along the arrows. For example, in 1999, the entire Web had 203 million webpages, and of these, 56 million were in the GSCC.

Even bigger than the GSCC is the "giant weakly connected component", or GWCC. This is the set of all nodes from which you can get to the GSCC by following a sequences of edges either forwards or backwards. In 1999, 186 million webpages were in the GWCC. That's 91% of all webpages!

We can also define the "giant in-component" or GIN to be the set of all nodes from which you can get into the GSCC by following edges forward. Similarly, the "giant out-component" or GOUT is the set of nodes that you can get to by going out of the GSCC, following edges forward. In 1999, both the GIN and the GOUT of the Web contained about 99 million webpages.

Besides these structures, there are also "tendrils" leading out of the GIN and into the GOUT. More precisely, "tendrils" consist of nodes in the GWCC but in neither the GIN nor the GOUT. In 1999, 44 million webpages lay in these tendrils.

Finally, there are a bunch of smaller components not reachable from the GSCC by edges pointing either forwards or backwards; in 1999 these accounted for 17 million webpages.

Of course, the main reason I'm interested in randomly evolving graphs is not because I surf the Web, but because I work on spin foam models of quantum gravity. Here the nodes and edges are labelled by spins, and instead of a probabilistic evolution rule one has a quantum-mechanical rule. So things are pretty different, though there are tantalizing similarities.

I gave a review of spin foam models and an introduction to the following new papers:

7) John Baez and J. Daniel Christensen, Positivity of spin foam amplitudes, available at gr-qc/0110044.

8) J. Daniel Christensen and Greg Egan, An efficient algorithm for the Riemannian 10j symbols, available at gr-qc/0110045.

The Riemannian 10j symbols are a function of ten spins that serves as the amplitude for a spin foam vertex in the Barrett-Crane model of Riemannian quantum gravity - by which I mean the theory where we do a real-time path integral over Riemannian metrics. This is different from so-called "Euclidean quantum gravity", where we do an imaginary-time path integral over Riemannian metrics. As far as I can tell, Riemannian quantum gravity is only important insofar as it's a useful warmup for Lorentzian quantum gravity.

In their paper, Christensen and Egan describe an algorithm that computes the Riemannian 10j symbols using O(j⁵) operations and O(j²) space, as well as an algorithm that uses O(j⁶) operations and a constant amount of space. This is in contrast to the most obvious methods, which use O(j⁹) operations and O(j²) or more space. Perhaps most importantly to the practical-minded among us, their paper includes a link to some code in C that implements this algorithm.

In our paper, Christensen and I show that the Riemannian 10j symbols are real, and that when they are nonzero, they are positive (resp. negative) when the sum of the ten spins is an integer (resp. half-integer). The proof is a nice exercise in spin network theory. We also show that for a closed spin foam of the type appearing in the Barrett-Crane model, the minus signs cancel when we take the product of Riemannian 10j symbols over all the spin foam vertices. It follows that in both the original Riemannian Barrett-Crane model, and also the modified version due to Perez and Rovelli, the amplitudes of spin foams are nonnegative.

This is interesting because, as Lee Smolin has often emphasized, it's hard to simulate spin foams on the computer unless the amplitudes are nonnegative. Nonnegative amplitudes allows us to use ideas from statistical mechanics, like the Metropolis algorithm. This is one reason lattice gauge theory people prefer imaginary-time path integrals to real-time ones. Of course, in lattice gauge theory, we can do Wick rotation to get real physics from imaginary-time path integrals. In quantum gravity, Wick rotation is more problematic, though Renate and others have considered situations where it's justified. It thus comes as a pleasant surprise to find that sometimes spin foam amplitudes are nonnegative without doing Wick rotation.

Of course, so far I've only been talking about the Riemannian Barrett-Crane model! Here the gauge group is Spin(4) = SU(2) × SU(2), and if you examine our proof, you'll see that the positivity result comes from the way this group "factors" into two copies of SU(2). We can't prove positivity of spin foam amplitudes in the more physical Lorentzian case, where the group is Spin(3,1) = SL(2,C).

However, even though we can't prove it, it may be true! Dan has written a number of programs which compute the Lorentzian 10j symbols, and while they are very slow and we haven't computed many values, all the values we've computed so far seem to be positive. We include the results we have so far in our paper.

In a paper that will come out later, "Partition function of the Riemannian Barrett-Crane model", by Dan Christensen, Tom Halford, David Tsang and myself, we'll discuss the qualitative behavior of various versions of the Riemannian Barrett-Crane model. In order to write this paper, we needed to numerically simulate the Barrett-Crane model using the Metropolis algorithm and the efficient algorithm for Riemannian 10j symbols.

Actually, in this conference there were lots of talks about different models of quantum gravity involving discrete random geometries. But right now I'll just discuss something called the IKKT matrix model. This was proposed in the following paper:

9) N. Ishibashi, H. Kawai, Y. Kitazawa and T. Tsuchiya, A large-N reduced model as superstring, Nucl. Phys. B498 (1997) 467-491. Also available as hep-th/9612115.

The idea is to provide something like a background-free formulation of type IIB string theory. But I don't understand how that's supposed to work yet, so my own attaction to this theory mainly comes from the fact that it's very simple and pretty. Let me describe it to you!

I'll assume you know that the Lagrangian for SU(N) Yang-Mills theory coupled to spinors looks like this:

tr(F ∧ ∗ F) + ψ^† D ψ

where F is the curvature of the gauge field, ψ is a spinor field transforming under some representation of SU(N), ψ^† is my feeble substitute for a ψ with a bar over it, and D is the covariant Dirac operator. If we write this out a bit more explicitly, it's

tr((dA + [A,A]) ∧ ∗(dA + [A,A]) + ψ^†i (d_a + A_a) Γ^a_ij ψ^j

where A is the gauge field. But now let's assume A and ψ are constant as functions on space, and that ψ transforms in the adjoint representation of su(N). This amounts to saying that A lies in su(N) ⊗ Rⁿ, where n is the dimension of spacetime, and that ψ lies in su(N) tensored with the space of spinors... where we use some sort of spinors suitable for n-dimensional spacetime. Then the above Lagrangian becomes

tr([A_a,A_b] [A^a,A^b]) + ψ^†i [A_a, Γ^aij ψ^j]

which is the Lagrangian for the IKKT model.

Now the idea is that as N → ∞, this sort of theory can reduce to string theory on some n-dimensional spacetime manifold... but not necessarily any fixed manifold.

It will be no surprise to readers of "week93" and "week104" that this model is supersymmetric when the spacetime dimension is 3, 4, 6, or 10. The reason is that in these dimensions both vectors and spinors have a nice description in terms of the real numbers, complex numbers, quaternions or octonions, respectively. The 10-dimensional octonionic version is the one that string theorists hope is related to the type IIB superstring. In this case, we can think of both A and ψ as big fat matrices of octonions!

There were a few different talks about the IKKT matrix model. John Wheater gave a talk about results saying that the path integral converges for this model in certain cases. In particular, it converges if n = 4, 6, or 10. For more details try this:

10) Peter Austing and John F. Wheater, Convergent Yang-Mills matrix theories, JHEP 0104 (2001) 019. Also available as hep-th/0103159.

Bengt Petersson spoke about computer simulations of the IKKT model:

11) Z. Burda, B. Petersson, J. Tabaczek, Geometry of reduced supersymmetric 4D Yang-Mills integrals, Nucl. Phys. B602 (2001) 399-409. Also available as hep-lat/0012001.

Also, Graziano Vernizzi spoke on work still in progress attempting to see the compactification of spacetime from 10 to 4 dimensions in superstring theory as a natural consequence of a matrix model.

For more on the IKKT model, try this:

12) A. Konechny and A. Schwarz, Introduction to M(atrix) theory and noncommutative geometry, available at hep-th/0012145.

There were a lot more talks, but on my way back home I started reading some papers about Tarski's "high school algebra problem", so now let me talk about that. This is more like mathematical logic than mathematical physics... at least at first. If you follow it through long enough, it turns out to be related to stuff like Feynman diagrams, but I doubt I'll have the energy to go that far this week.

So:

Once upon a time, the logician Tarski posed the following question. Are there any identities involving addition, multiplication, exponentiation and the number 1 that don't follow from the identities we all learned in high school? In case you forgot, these are:

x + y = y + x        (x + y) + z = x + (y + z) 

xy = yx              (xy)z = x(yz)

1x = x

x^1 = x               1^x = 1

x(y + z) = xy + xz 

x^(y + z) = x^y x^z  (xy)^z = x^z y^z    x^(yz) = (x^y)^z

Remarkably, in 1981 it turned out the answer is YES:

13) A. J. Wilkie, On exponentiation - a solution to Tarski's high school algebra problem, to appear in Quaderni di Matematica. Also available at http://www.maths.ox.ac.uk/~wilkie/

Here is Wilkie's counterexample:

[(x + 1)^x + (x^2 + x + 1)^x]^y  [(x^3 + 1)^y + (x^4 + x^2 + 1)^y]^x =
[(x + 1)^y + (x^2 + x + 1)^y]^x  [(x^3 + 1)^x + (x^4 + x^2 + 1)^x]^y 

Wilkie's original proof was rather subtle, but in 1985 Gurevic gave a more simple-minded proof: he constructed a finite set equipped with addition, multiplication, exponentiation and 1 satisfying the high school algebra axioms but not Wilkie's identity. This clearly shows that the former don't imply the latter! His counterexample had 59 elements:

14) R. Gurevic, Equational theory of positive numbers with exponentiation, Proc. Amer. Math. Soc. 94 (1985), 135-141.

Later, various mathematicians enjoyed cutting down the number of elements in this counterexample. As far as I can tell, the current record-holder is Marcel Jackson, who constructed one with only 14 elements. He also showed that none exists with fewer than 8 elements:

15) Marcel G. Jackson, A note on HSI-algebras and counterexamples to Wilkie's identity, Algebra Universalis 36 (1996), 528-535. Also available at http://www.latrobe.edu.au/mathstats/Staff/Marcel/details/publications.html

I have no idea what these small counterexamples are good for, though Jackson proves some nice things in the process of studying them.

More important, in my opinion, is a 1990 result of Gurevic: no finite set of axioms in first-order logic is sufficient to prove all the identities involving addition, multiplication, exponentiation and 1 that hold for the positive natural numbers. You can find this here:

16) R. Gurevic, Equational theory of positive numbers with exponentiation is not finitely axiomatizable, Ann. Pure. Appl. Logic 49 (1990), 1-30.

In other words, Wilkie's identity is but one of an infinite set of logically independent axioms of this type!

But the real fun starts when we categorify Tarski's high school algebra problem. I learned about this from Marcelo Fiore, a computer scientist whom I met in Cambridge this summer. The idea here is to realize that the high school identities all hold as isomorphisms between finite sets if we interpret addition as disjoint union, multiplication as Cartesian product, x^y as the set of functions from the finite set y to the finite set x, and 1 as your favorite one-element set. The point here that the set of natural numbers is just a dumbed-down version of the category of finite sets, with all these arithmetic operations coming from things we can do with finite sets. I explained this in "week121".

From this viewpoint it's very natural to include some extra axioms involving 0, which corresponds to the empty set:

0 + x = x        

0x = 0     

x^0 = 1

The reason why it's so nice to include 0 is that then the high school identities correspond closely to what holds in any "biCartesian closed category" - a good example being the category of finite sets. A Cartesian category is one with binary products and a terminal object; these act like "multiplication" and "1". In a Cartesian closed category we also require that the operation of taking the product with any object has a right adjoint; this gives "exponentiation". Finally, in a biCartesian closed category we also have binary coproducts and an initial object, which act like "addition" and "0", and we require that products distribute over coproducts.

There are lots of examples of biCartesian closed categories: for example, the category of finite sets, or sets, or sets on which some group acts, or more generally presheaves on any category, or still more generally, any topos!

Anyway, Fiore has solved the following categorified version of Tarski's high school algebra problem, posed by Roberto di Cosmo: are there any natural isomorphisms in the category of finite sets between expressions built from addition, multiplication, exponentiation, 0 and 1 that don't hold in a general biCartesian closed category? I'm posing this a bit vaguely, so I hope you can guess what I mean. Anyway, the answer is again YES, and a similar sort of counterexample does this job.

To tackle this problem it's useful to consider the free biCartesian closed category on some set of objects, because this has the fewest isomorphisms. Now, the real reason I'm interested in this stuff is that James Dolan and Toby Bartels have been thinking about various similar categories, like the free Cartesian closed category on one object, or the free symmetric monoidal closed category on one object, or the free symmetric monoidal compact category on one object... and the last-mentioned of these is closely related to the theory of Feynman diagrams!

But alas, just as I suspected, I don't have the energy to go into this now. So I'll stop here, hopefully leaving you more tantalized than baffled.

(Thanks go to Michael Barr, Noam Elkies, Dave Rusin and Bruce Smith for catching mistakes in the original error-ridden version of this issue.)

Postscript - A friend of mine interested in nanotechnology made the following comments on Wil McCarthy's article:

From your comments I can't tell if you were "fooled" by his misleading impression in the article that (1) the specific physical atoms of our periodic table could be "found" analogously in quantum dots containing the corresponding numbers of electrons; or the weaker (but still false) (2) there could be a "periodic table of quantum-dot artificial atoms" indexed by their number of electrons. (2) is the most obviously false, and he even says why in the article - the shape of the dot (and for that matter the material it's made of) also influence its properties. But, basically for exactly this reason, (1) is also false - there's no reason to expect any quantum dot and number of electrons in it to be able to imitate a specific kind of physical atom.

So it's misleading to say that a material could be "switched from lead to gold". It could not be exactly "lead", and it could not be exactly "gold", and it could not even be in states which would justify making an analogy to those specific elements, unless you carefully selected the properties you wanted to compare -- e.g. color. But selecting color you might say "it's like gold", whereas selecting conductivity you might choose a different element (or something different from any element) to compare it to.

However, effects almost as interesting might be true (though I don't know enough to judge critically whether they really might be true), e.g. a material whose various physical properties could be quickly changed over wide ranges, "programmed" in various uncommon useful combinations, and reconfigured in tiny detail. It seems almost certainly true that an advanced nanotech would include important technological uses for these kinds of effects. Whether they can be useful in these "chemistry-like ways" before we have advanced nanotech (for building the dots precisely) is doubtful to me - their properties are likely to be highly dependent on their precise shape and composition, which I doubt we can control well enough without building them atom-by-atom. (However, they'll probably be quite useful in other ways, not analogous to "atoms", which depend much less on their precise shape & composition. I think this has already happened.)

It's too bad he gave the false impressions in the article, since it obscures the true and amazing stuff - it makes me unclear on how much of what he says is actually plausible.

One other thing he implied, which is false, is that "regular" nanotech couldn't give us anything like "programmable matter". In fact, if you really want a surface that switches from lead to gold at the flick of a switch, just make lots of little cubes or plates with gold on one side and lead on the other side, and have them all get turned over by little motors when you flick the switch.

This kind of "mechanical reconfiguration" method (generalized/extended a lot) could be fast enough to let big buildings change shape faster than water can flow, and with feasible expenditure of energy and generation of waste heat. So the main thing added to this by the possibility of "artificial atoms in quantum dots" would be a wider variety of electronic/optical/magnetic materials properties (I doubt the mechanical properties will be very much affected), and the ability to switch those in picoseconds (that's a guess) rather than merely milliseconds, and to do so for much less energy.

I.e. mainly important for technological uses rather than something that has a biq qualitative effect on "human experience", which will already include all effects that he listed, just from "regular" nanotech.

> ...But even so, I believe there will be some cool technological 
> applications of quantum dots.

Yep!

Also, here are some comments by Noam Elkies about the number 24:

John Baez wrote:

[....]

>Gannon's article is full of juicy mathematical tidbits and pieces
>of wisdom.  He even gives a new explanation of why the number 24
>is so important throughout mathematics and string theory.
>If x2 = 1 mod n, then x does not divide n... and 24
>is the largest integer for which the converse holds!

This cannot be right: for any n, the only factors x|n such that x² is 1 mod n are x=1 and x=-1. You must mean that n|24 if and only if x²-1 is a multiple of n for every integer x that's coprime to n. But is this connection really new? I remember observing this some time back, and can't believe I was the first either...

>Alas, Gannon does not explain how this relates to the other
>magic properties of this number, some of which are listed in "week124".
>Does anyone see the connection?

Here's one not-immediately-obvious consequence. Consider the group Γ₀(n²), consisting of 2×2 integer matrices of determinant 1 whose bottom left entry is a multiple of n². When is the matrix

       [ 1  1/n ]
T(n) = [        ]
       [ 0   1  ]

in the normalizer of this group? The conjugate of [a, b; n² c, d] by this matrix has integer diagonal entries and bottom left entry n² c; so it's in Γ₀(n²) if and only if the top right entry is an integer. Well, the top right entry is b-c + (d-a)/n. This is an integer provided d is congruent to a mod n. But all that restricts (a,d) mod n is the condition that ad-n²bc=1, and thus that ad=1 mod n. So, this should entail a=d mod n, which it does if and only if every integer coprime to n is its own multiplicative inverse mod n !

So, the integers n for which this holds are precisely those for which the normalizer of Γ₀(n²) contains T(n).

Another way to say this is: conjugate Γ₀(n²) by the matrix

  [ n   0  ]
  [        ]
  [ 0  1/n ]

This yields all integer matrices of determinant 1 whose off-diagonal matrices are multiples of n. Reducing mod n, we get the group of scalar matrices if and only if each unit in Z/nZ is a square root of 1 -- in which case we have a normal subgroup of SL₂(Z) [the group of 2×2 integer matrices of determinant 1], so in particular the corresponding conjugate

  [ 1   1 ]
  [       ]
  [ 0   1 ]

of T(n) is in the normalizer.

What has all this to do with moonshine? I'm no moonshine expert, so I can't say for sure; but moonshine certainly involves coefficients of modular forms and functions for congruence subgroups of SL₂(Z). If T(n) is in the normalizer of Γ₀(n²) then T(n) acts on the spaces of modular forms/functions by linear transformations whose n-th power is the identity (since T(n)ⁿ is in Γ₀(n²)). The eigenspaces of these transformations are the modular forms/functions whose coefficients are supported on arithmetic progressions mod n. So, we get to isolate the different arithmetic progressions mod n precisely when n satisfies the n|x²-1 condition. This should explain the special role played by these integers n, which as we know are 24 and its factors.

--Noam D. Elkies

And here is some more by Noam Elkies:

John Baez wrote:
 
>[...] He then uses this to explain why even self-dual lattices occur
>only in dimensions that are multiples of 8, which is nice, but he
>doesn't connect up with any appearances of the number 24.  
>Your remarks go much further in this direction - thanks!  

Glad to be of help. I mention the use of modular forms to explain this divisibility by 8 in my paper "A characterization of the Zⁿ lattice" (Math Research Letters 2 (1995), 321-6 = math.NT/9906019, and again in the first part of my expository article "Lattices, Linear Codes, and Invariants" (AMS Notices 27 (2000), pages 1238-1245 = http://www.ams.org/notices/200010/fea-elkies-1.pdf; see the footnote on page 1243).

>To me, one basic reason for the appearance of the number 24
>in the theory of modular forms is the fact that of all lattices
>in the plane, the square one and the "equilateral triangle" one 
>have more symmetry - 4-fold symmetry and 6-fold symmetry,
>respectively.  It's related to the fact that the abelianization
>of SL(2,Z) has 12 elements.  But I don't see an immediate connection
>between these simple things and the above number-theoretic property 
>of 24.

I don't see a complete explanation either. However, there is this:

The group of units in Z/nZ is known to be the Galois group over Q of the n-th cyclotomic field (the field obtained from Q by adjoining the n-th roots of unity).

The condition: (x,n)=1 iff n|x²-1 is equivalent to the requirement that every element of this group be its own inverse, and thus that this group be isomorphic to (Z/2Z)^r.

By Galois theory, this is equivalent to requirement that the n-th cyclotomic field be the compositum of r quadratic extensions. For instance, the 24th cyclotomic field is the compositum of Q(i), Q(√-3), and Q(√2).

Now if a lattice L in C has extra symmetries, then its ring of complex endomorphisms (the complex numbers z such that zL is contained in L) is an imaginary quadratic field generated by these symmetries. It is thus one of the two cyclotomic fields of degree 2 over Q.

So this explains at least why 4 and 6 (as in fourfold and sixfold symmetry) are factors of the number 24. It doesn't explain 24 entirely, because 12 suffices to get both 4 and 6. But then 12 is also a good number for this kind of game; see Poonen and Rodriguez-Villegas's paper (www.math.berkeley.edu/~poonen/papers/lattice12.ps) on "Lattice polygons and the number 12".

NDE

Finally, Marcelo Fiore tells me that there are some extra axioms for 0 which automatically arise when you decategorify a biCartesian closed category, for example:

    x ⋅ 0^x = 0          (logically: x and not x is false)
 
   0^(0^(0^x)) = 0^x     (logically: not not not x iff not x)