Ulam's packing conjecture, named for Stanisław Ulam, holds that the highest possible packing density of identical convex solids in three-dimensional Euclidean space is achieved by no convex body better than the sphere, so the ball is the convex solid that forces the largest fraction of space to remain empty in its optimal packing. It is related to the Kepler conjecture, whose solution establishes that identical balls must leave about 25.95 percent of space empty.
Facts
StatementUlam asked whether any three dimensional convex body has a lower maximum packing density than the sphere, that is, whether some convex solid forces even more empty space in its optimal packing than the sphere does. 1 Progress Toward ResolutionNumerical experiments on many convex solids have only produced packings less space efficient than sphere packing, ruling those shapes out as counterexamples, but an infinite space of untested shapes remains untested, so the conjecture stays open. 1 Partially Attested
Proposed Year1972 is the year Ulam communicated the conjecture to Martin Gardner, the earliest dated record in the source; the source does not give the year Ulam first conceived it, and says Gardner attributed it to Ulam posthumously. Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Ulam's Packing Conjecture (Wikipedia)
Posed By
Source Ulam's Packing Conjecture (Wikipedia)
Sources
1. Ulam's Packing Conjecture (Wikipedia)
Lead section
Ulam's packing conjecture, named for Stanisław Ulam, is a conjecture about the highest possible packing density of identical convex solids in three-dimensional Euclidean space. The conjecture says that the optimal density for packing congruent spheres is smaller than that for any other convex body.
Current status section
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, 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.
- Lead section
View the Source2. Ulam's Packing Conjecture (Wikipedia)
Wikimedia FoundationLead section
the ball is the convex solid which forces the largest fraction of space to remain empty in its optimal packing structure
Origin section
Ulam communicated this conjecture to him in 1972
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