J. Kepler

The packing will be the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container.

D. Hilbert

How can one arrange most densely in space an infinite number of equal solids of given form, e.g., spheres with given radii..., that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible? (Hilbert's Problem 18, part C, 1900).

S. Singh

Of all the problems likely to replace Fermat's Last Theorem as the greatest unsolved problem in mathematics the best candidate is Kepler's sphere-packing problem. - S. Singh in Fermat's Last Theorem.

R. Kargon

Hariot advised Kepler to abstract himself mathematically into an atom in order to enter `Nature's house'. In his reply of 2 August 1607, Kepler declined to follow Harriot, ad atomos et vacua. Kepler preferred to think of the reflection-refraction problem in terms of the union of two opposing qualities--transparence and opacity. Hariot was surprised. `If those assumptions and reasons satisfy you, I am amazed.'

H.S.M. Coxeter

The densest lattice-packing is not necessarily the densest packing. A first suspicion in this direction arises from the existence of equally dense non-lattice packings: the hexagonal close packing and also several hybrids.

C. A. Rogers

While many mathematicians believe, and all physicists know, that the density cannot exceed pi/sqrt(18)=0.74048..., the best properly established bound known [before 1958] seems to have been the bound 0.828,... obtained by Rankin.

D. J. Muder

It's one of those problems that tells us that we are not as smart as we think we are.

Jean-Michel Kantor

Hilbert's text gives the impression that he did not anticipate the success and the developments this problem would have.

W. Kahan

The Directed roundings can be used to implement Interval Arithmetic, which is a scheme that approximates every variable not by one value of unknown reliability but by two that are guaranteed to straddle the ideal value. This scheme is not so popular in the U.S.A. as it is in parts of Europe, where some people distrust computers.

C. A. Rogers

Little precise is known about the packing density in three or more dimensions.

L. Fejes Tóth

Thus it seems that the problem can be reduced to the determination of the minimum of a function of a finite number of variables, providing a programme realizable in principle. In view of the intricacy of this function we are far from attempting to determine the exact minimum. But, mindful of the rapid development of our computers, it is imaginable that the minimum may be approximated with great exactitude.

J. Milnor

For 2-dimensional disks this problem has been solved by Thue and Fejes Toth,.. However, the corresponding problem in 3 dimensions remains unsolved. This is a scandalous situation since the (presumably) correct answer has been known since the time of Gauss.

J. Milnor

All that is missing is a proof.

S. Singh

It looks simple at first sight, but reveals its subtle horrors to those who try to solve it.