In the previous chapters, we have repeatedly referred to the exponential convergence rate of spectral methods for analytic functions. This is discussed in more detail in Section 4.1. When functions are not smooth, PS theory is much less clear. An approximation can appear very good in one norm and, at the same time, very bad in another. As illustrated in Section 4.2, PS performance can also be very impressive in many cases that are “theoretically questionable” – this is exploited in most major PS applications. Sections 4.3–4.5 describe differentiation matrices in more detail, their influence on time stepping procedures, and linear stability conditions. A fundamentally different kind of instability, specific to nonlinear equations, is discussed in Section 4.6. Very particular distributions of the nodes yield spectacular accuracies for Gaussian quadrature formulas. PS methods are often based on such formulas, presumably with the hope of obtaining a correspondingly enhanced accuracy when approximating differential equations. In the examples of Section 4.7, we see little evidence for this.

Smoothness of a function is a rather vague concept. Increasingly severe requirements include:

• a finite number of continuous derivatives;

• infinitely many derivatives; and

• analyticity – allowing continuation as a differentiate complex function away from the real axis.

In the limit of N (number of nodes or gridpoints) tending to infinity, these cases give different asymptotic convergence rates for PS methods. In the first case, the rate becomes polynomial with the power corresponding to the number of derivatives that are available.

The main feature of this appendix is Table A6.1 which lists the twenty-six sporadic simple groups. Most of the information is taken directly from the corresponding tables in [37, pp. 172-176] and [65, p. 709].

As the table clearly shows, about the only items in complete agreement are the group orders. We have taken the dates of discovery from [65] as they indicate dates of initial discovery and not necessarily of the completed proof. The superscripts at the end of the discoverer(s’) name(s) are used whenever there was a discrepancy. These correspond to the cited references.

Table A6.1 appears on pages 212–215.

Section 1. An Introduction to Sphere Packing

In Euclidean n-space, En, how may disjoint, open, congruent n-spheres be located to maximize the fraction of the volume of En that the n-spheres cover? That is the sphere packing problem, which goes back to a book review that Gauss wrote in 1831, in which he pointed out that a problem concerning the minimal nonzero value assumed by a positive definite quadratic form in n variables, first considered by Lagrange in 1773, could be translated into a problem on packing spheres (cf. C. A. Rogers [105, pp. 1, 106]).

Though there might seem to be no connection between coding theory and sphere packing, John Leech in [72] and [74] used results in coding theory to obtain a packing of E24 that was much denser than any previously known. The present chapter will describe his packing and discuss how he happened to bridge the gap, so to speak, from codes to sphere packings.

The (23, 12) Golay code can be extended to a (24, 12) code by adding a 0 or 1 to each codeword so that all code-words have even weight. This (24, 12) Golay code thus has a minimum codeword distance of eight.

Back in 1947 Richard W. Hamming had access to a computer only on weekends. Some three decades later he recalled his frustration over its perverse behavior:

That question initiated the development of error-correcting codes. The offending computer, a mechanical relay model at Bell Telephone Laboratories, always came to an angry stop and switched to the next program whenever it detected an error. This behavior impelled Hamming, a pure mathematician with an applied bent, to devise the first error-correcting code.

We shall follow the devious trail that wends its way through a quarter-century of mathematics, starting with Hamming's work, which led almost immediately to that of M. J. E. Golay. The latter sparked, some twelve years later, giant steps in the packing of congruent spheres by John Leech, which, in turn, branched off through the work of J. H. Conway, into the field of simple groups. By tracing some of the twists, turns, switchbacks and dead ends of this path, we hope to provide a small window on the history of mathematics of the twentieth century. How historians ultimately will treat this golden age of mathematical creativity we cannot even guess.

Section 1. An Introduction to Coding

Richard W. Hamming's encounter with the Bell Telephone Laboratories' mechanical relay computer in 1947 (quoted in the Preface) initiated what has come to be known as coding theory. In this chapter we will trace these origins of coding theory and develop enough of the theory itself to prepare for Leech's work in sphere packing which appears in Chapter 2.

Communication is imperfect. Even if a message is accurately stated, it may be garbled during transmission; and the consequences of a mistake in the interpretation of a financial, diplomatic, military or other message may be unfortunate. (Hamming's work was goaded by mistakes which occurred internally in a computer.) As a result, when a message travels from an information source to a destination, both the sender and the receiver would like assurance that the received message either is free of errors or, if it contains errors, those errors will be detectable. Ideally, in case an error is detected, the receiver would like to be able to correct it and recover the original message.

Our interest will lie mainly in the transmission of strings of 0's and 1's of length n, which are called (binary) n-blocks. This may, at first glance, seem an artificial restriction. But in fact the entire twenty-six letters of the alphabet can be coded using 5-blocks since there are thirty-two arrangements of five 0's and 1's.

The main feature of this Appendix is Table A1.1, a list of the densest known packings in En for n = 1 through 24, and for a few selected n larger than 24. The Table is essentially that in [83] updated by material from [112], [113] and [95]. It differs from that in [83], however, in that density, instead of center density is used. (Recall that the density, ρ, of a packing in En is the fraction of En within the spheres of the packing, while the center density,δ, is ρ divided by Vn, the volume of a sphere in the packing.) The latter is the number of centers of unit spheres per unit n-dimensional volume.

Several other comments on Table A1.1 are in order.

1. Under the column marked “Type,” B indicates that both a lattice and a nonlattice packing with these parameters are known. L indicates that at present only a lattice packing is known, and N that only a nonlattice packing is known. A indicates a local arrangement of spheres touching one sphere.

2. N. J. A. Sloane kindly provided the sources [112], [113] and [95]. Specifically, the packing P10c in E10 appears in [113, p. 31], the packings P11c in E11 and one in E36 with no designation appear in [112, p. 118], and new bounds for contact numbers appear in [95].