Higher Dimensions and Several Variables We have already seen that the study of polynomial

equations becomes much more complicated when one looks not just at single equations in one variable, but at systems of equations in several variables. Sim- ilarly, we have seen thatpartial differential equa- tions[I.3 §5.4], which can be thought of as differen- tial equations involving several variables, are typically much more difficult to analyze than ordinary differen- tial equations, that is, differential equations in just one variable. These are two notable examples of a process that has generated many of the most important prob- lems and results in mathematics, particularly over the last century or so: the process of generalization from one variable to several variables.

Suppose one has an equation that involves three real variables, x, y, and z. It is often useful to think of

Figure 1The densest possible packing of circles in the plane.

the triple(x, y, z)as an object in its own right, rather than as a collection of three numbers. Furthermore, this object has a natural interpretation: it represents a point in three-dimensional space. This geometrical interpretation is important, and goes a long way toward explaining why extensions of definitions and theorems from one variable to several variables are so interest- ing. If we generalize a piece of algebra from one vari- able to several variables, we can also think of what we are doing as generalizing from a one-dimensional set- ting to a higher-dimensional setting. This idea leads to many links between algebra and geometry, allowing techniques from one area to be used to great effect in the other.

4 Discovering Patterns

Suppose that you wish to fill the plane as densely as possible with nonoverlapping circles of radius 1. How should you do it? This question is an example of a so- calledpacking problem. The answer is known, and it is what one might expect: you should arrange the cir- cles so that their centers form a triangular lattice, as shown in figure 1. In three dimensions a similar result is true, but much harder to prove: until recently it was a famous open problem known as the Kepler conjec- ture. Several mathematicians wrongly claimed to have solved it, but in 1998 a long and complicated solution, obtained with the help of a computer, was announced by Thomas Hales, and although his solution has proved very hard to check, the consensus is that it is probably correct.

Questions about packing of spheres can be asked in any number of dimensions, but they become harder and harder as the dimension increases. Indeed, it is likely that the best density for a ninety-seven-dimen- sional packing, say, will never be known. Experience with similar problems suggests that the best arrange- ment will almost certainly not have a simple structure such as one sees in two dimensions, so that the only

method for finding it would be a “brute-force search”

of some kind. However, to search for the best possible complicated structure is not feasible: even if one could somehow reduce the search to finitely many possibili- ties, there would be far more of them than one could feasibly check.

When a problem looks too difficult to solve, one should not give up completely. A much more produc- tive reaction is to formulate related but more approach- able questions. In this case, instead of trying to discover the very best packing, one can simply see how dense a packing one can find. Here is a sketch of an argument that gives a goodish packing inndimensions, whenn is large. One begins by taking amaximal packing: that is, one simply picks sphere after sphere until it is no longer possible to pick another one without it overlap- ping one of the spheres already chosen. Now letxbe any point in_Rⁿ. Then there must be a sphere in our collection such that the distance betweenxand its cen- ter is less than 2, since otherwise we could take a unit sphere aboutx and it would not overlap any of the other spheres. Therefore, if we take all the spheres in the collection and expand them by a factor of 2, then we cover all of_Rⁿ. Since expanding ann-dimensional sphere by a factor of 2 increases its (n-dimensional) volume by a factor of 2ⁿ, the proportion of_Rⁿcovered by the unexpanded spheres must be at least 2⁻ⁿ.

Notice that in the above argument we learned nothing at all about the nature of the arrangements of spheres with density 2⁻ⁿ. All we did was take a maximal pack- ing, and that can be done in a very haphazard way. This is in marked contrast with the approach that worked in two dimensions, where we defined a specific pattern of circles.

This contrast pervades all of mathematics. For some problems, the best approach is to build a highly struc- tured pattern that has the properties you need, while for others—usually problems for which there is no hope of obtaining an exact answer—it is better to look for less specific arrangements. “Highly structured” in this context often means “possessing a high degree of symmetry.”

The triangular lattice is a rather simple pattern, but some highly structured patterns are much more com- plicated, and much more of a surprise when they are discovered. A notable example occurs in packing prob- lems. By and large, the higher the dimension you are working in, the more difficult it is to find good pat- terns, but an exception to this general rule occurs at twenty-four dimensions. Here, there is a remarkable

construction, known as theLeech lattice, which gives rise to a miraculously dense packing. Formally, alattice in_Rⁿis a subsetΛwith the following three properties.

(i) Ifxandybelong toΛ, then so dox+yandx−y.

(ii) Ifxbelongs toΛ, thenxisisolated. That is, there is somed >0 such that the distance betweenx and any other point ofΛis at leastd.

(iii) Λ is not contained in any (n−1)-dimensional subspace of_Rⁿ.

A good example of a lattice is the set_Zⁿof all points in_Rⁿwith integer coordinates. If one is searching for a dense packing, then it is a good idea to look at lat- tices, since if you know that every nonzero point in a lattice has distance at leastdfrom 0, then you know thatanytwo points have distance at leastdfrom each other. This is because the distance betweenxandyis the same as the distance between 0 andy−x, both of which lie in the lattice ifxandy do. Thus, instead of having to look at the whole lattice, one can get away with looking at a small portion around 0.

In twenty-four dimensions it can be shown that there is a latticeΛwith the following additional properties, and that it is unique, in the sense that any other lattice with those properties is just a rotation of the first one.

(iv) There is a 24×24 matrixM withdeterminant [III.15] equal to 1 such thatΛconsists of all integer combinations of the columns ofM.

(v) Ifvis a point inΛ, then the square of the distance from 0 tovis an even integer.

(vi) The nonzero vector nearest to 0 is at distance 2.

Thus, the balls of radius 1 about the points inΛ form a packing of_R²⁴.

The nonzero vector nearest to 0 is far from unique: in fact there are 196 560 of them, which is a remarkably large number considering that these points must all be at distance at least 2 from each other.

The Leech lattice also has an extraordinary degree of symmetry. To be precise, it has 8 315 553 613 086 720 000 rotational symmetries. (This number equals 2²²·3⁹·5⁴·7²·11·13·23.) If you take the quo- tient[I.3 §3.3] of its symmetry group by the subgroup consisting of the identity and minus the identity, then you obtain theConway groupCo1, which is one of the famous sporadicsimple groups[V.7]. The existence of so many symmetries makes it easier still to determine the smallest distance from 0 of any nonzero point of the lattice, since once you have checked one distance

you have automatically checked lots of others (just as, in the triangular lattice, the six-fold rotational symme- try tells us that the distances from 0 to its six neighbors are all the same).

These facts about the Leech lattice illustrate a general principle of mathematical research: often, if a mathe- matical construction has one remarkable property, it will have others as well. In particular, a high degree of symmetry will often be related to other interesting fea- tures. So, although it is a surprise that the Leech lattice exists at all, it is not as surprising when one then dis- covers that it gives an extremely dense packing of_R²⁴. In fact, it was shown in 2004 by Henry Cohn and Abhi- nav Kumar that it gives the densest possible packing of spheres in twenty-four-dimensional space, at least among all packings derived from lattices. It is probably the densest packing of any kind, but this has not yet been proved.

5 Explaining Apparent Coincidences The largest of all the sporadic finite simple groups is called theMonster group. Its name is partly explained by the fact that it has 2⁴⁶·3²⁰·5⁹·7⁶·11²·13³·17· 19·23·29·31·41·47·59·71 elements. How can one hope to understand a group of this size?

One of the best ways is to show that it is a group of symmetries of some other mathematical object (see the article onrepresentation theory[IV.9] for much more on this theme), and the smaller that object is, the better. We have just seen that another large sporadic group, the Conway group Co₁, is closely related to the symmetry group of the Leech lattice. Might there be a lattice that played a similar role for the Monster group?

It is not hard to show that there will be at leastsome lattice that works, but more challenging is to find one of small dimension. It has been shown that the smallest possible dimension that can be used is 196 883.

Now let us turn to a different branch of mathemat- ics. If you look at the article aboutalgebraic numbers [IV.1 §8] you will see a definition of a function j(z), called theelliptic modular function, of central impor- tance in algebraic number theory. It is given as the sum of a series that starts

j(z)=e^−2πiz+744+196 884e^2πiz

+21 493 760e^4πiz+864 299 970e^6πiz+ · · ·. Rather intriguingly, the coefficient of e^2πizin this series is 196 884, one more than the smallest possible dimen- sion of a lattice that has the Monster group as its group of symmetries.

It is not obvious how seriously we should take this observation, and when it was first made by John McKay opinions differed about it. Some believed that it was probably just a coincidence, since the two areas seemed to be so different and unconnected. Others took the attitude that the functionj(z)and the Monster group are so important in their respective areas, and the num- ber 196 883 so large, that the surprising numerical fact was probably pointing to a deep connection that had not yet been uncovered.

It turned out that the second view was correct. After studying the coefficients in the series forj(z), McKay and John Thompson were led to a conjecture that related them all (and not just 196 884) to the Mon- ster group. This conjecture was extended by John Con- way and Simon Norton, who formulated the “Monstrous Moonshine” conjecture, which was eventually proved by Richard Borcherds in 1992. (The word “moonshine”

reflects the initial disbelief that there would be a seri- ous relationship between the Monster group and the j-function.)

In order to prove the conjecture, Borcherds intro- duced a new algebraic structure, which he called aver- tex algebra[IV.17]. And to analyze vertex algebras, he used results fromstring theory[IV.17 §2]. In other words, he explained the connection between two very different-looking areas of pure mathematics with the help of concepts from theoretical physics.

This example demonstrates in an extreme way anoth- er general principle of mathematical research: if you can obtain the same series of numbers (or the same structure of a more general kind) from two different mathematical sources, then those sources are prob- ably not as different as they seem. Moreover, if you can find one deep connection, you will probably be led to others. There are many other examples where two completely different calculations give the same answer, and many of them remain unexplained. This phenomenon results in some of the most difficult and fascinating unsolved problems in mathematics. (See the introduction tomirror symmetry[IV.16] for another example.)

Interestingly, thej-function leads to a second famous mathematical “coincidence.” There may not seem to be anything special about the number e^π√163, but here is the beginning of its decimal expansion:

e^π√163

=262 537 412 640 768 743.99999999999925. . . ,

which is astonishingly close to an integer. Again it is initially tempting to dismiss this as a coincidence, but one should think twice before yielding to the tempta- tion. After all, there are not all that many numbers that can be defined as simply as e^π√163, and each one has a probability of less than one in a million million of being as close to an integer as e^π√163is. In fact, it is not a coincidence at all: for an explanation seealgebraic numbers[IV.1 §8].

6 Counting and Measuring

How many rotational symmetries are there of a regular icosahedron? Here is one way to work it out. Choose a vertexvof the icosahedron and letv be one of its neighbors. An icosahedron has twelve vertices, so there are twelve places wherevcould end up after the rota- tion. Once we know wherevgoes, there are five possi- bilities forv (since each vertex has five neighbors and v must still be a neighbor ofvafter the rotation). Once we have determined wherevandv go, there is no fur- ther choice we can make, so the number of rotational symmetries is 5×12=60.

This is a simple example of acounting argument, that is, an answer to a question that begins “How many.”

However, the word “argument” is at least as important as the word “counting,” since we do not put all the sym- metries in a row and say “one, two, three,. . ., sixty,” as we might if we were counting in real life. What we do instead is come up with a reason for the number of rotational symmetries being 5×12. At the end of the process, we understand more about those symmetries than merely how many there are. Indeed, it is possible to go further and show that the group of rotations of the icosahedron isA₅, thealternating group[III.68]

on five elements.