Ulam's packing conjecture, named for Stanislaw Ulam, is a conjecture about the highest possible packing density of identical convex solids in three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

. The conjecture says that the optimal density for packing congruent spheres is smaller than that for any other convex body. That is, according to the conjecture, the ball is the convex solid which forces the largest fraction of space to remain empty in its optimal packing structure. This conjecture is therefore related to the

Kepler conjecture The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling s ...

about

sphere packing In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing p ...

. Since the solution to the Kepler conjecture establishes that identical balls must leave ≈25.95% of the space empty, Ulam's conjecture is equivalent to the statement that no other convex solid forces that much space to be left empty.

Origin

This conjecture was attributed posthumously to Ulam by

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...

, who remarks in a postscript added to one of his ''

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'' columns that Ulam communicated this conjecture to him in 1972. Though the original reference to the conjecture states only that Ulam "suspected" the ball to be the worst case for packing, the statement has been subsequently taken as a conjecture.

Supporting arguments

Numerical experiments with a large variety of convex solids have resulted in each case in the construction of packings that leave less empty space than is left by

close-packing of equal spheres In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...

, and so many solids have been ruled out as counterexamples of Ulam's conjecture. Nevertheless, there is an infinite space of possible shapes that have not been ruled out. Yoav Kallus has shown that at least among point-symmetric bodies, the ball constitutes a local maximum of the fraction of empty space forced. That is, any point-symmetric solid that does not deviate too much from a ball can be packed with greater efficiency than can balls.

Analogs in other dimensions

The analog of Ulam's packing conjecture in two dimensions would say that no convex shape forces more than ≈9.31% of the plane to remain uncovered, since that is the fraction of empty space left uncovered in the densest packing of disks. However, the regular octagon and smoothed octagon give counter-examples. It is conjectured that regular heptagons force the largest fraction of the plane to remain uncovered. In dimensions above four (excluding 8 and 24), the situation is complicated by the fact that the analogs of the

remain open.

References

{{Reflist Packing problems Conjectures Spheres Unsolved problems in geometry