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Ulam's packing conjecture

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.

Version
v1 · 2026-09-28 · History
Domain-specific #
12686
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Discrete Geometry → Mathematics

Core Idea

Ulam's packing conjecture is treated here as the recurring discrete geometry identity summarized by this source-grounded definition: 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. 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.

Scope of Application

  • Origin. This conjecture was attributed posthumously to Ulam by Martin Gardner, who remarks in a postscript added to one of his Mathematical Games columns that Ulam communicated this conjecture to him in.

  • Origin. 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. Nevertheless, there is an infinite space of possible shapes that have not been ruled out.

  • Supporting arguments. Yoav Kallus has shown that at least among point-symmetric bodies, the ball constitutes a local maximum of the fraction of empty space forced.

  • Supporting arguments. That is, any point-symmetric solid that does not deviate too much from a ball can be packed with greater efficiency than can balls.

Clarity

A clear use of Ulam's packing conjecture names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is 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.

Manages Complexity

Ulam's packing conjecture compresses multiple discrete geometry details into a stable diagnostic relation. The source shows both the central mechanism—in dimensions above four (excluding 8 and 24), the situation is complicated by the fact that the analogs of the Kepler conjecture remain open.—and the practical consequence—yoav Kallus has shown that at least among point-symmetric bodies, the ball constitutes a local maximum of the fraction of empty space.

Abstract Reasoning

  1. Type the carrier. Identify the discrete geometry entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: 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.
  3. Check operation and conditions. 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.

Knowledge Transfer

Within the home domain. Knowledge about Ulam's packing conjecture transfers literally when a new case preserves the same carrier type, relation, and recognition test. This conjecture was attributed posthumously to Ulam by Martin Gardner, who remarks in a postscript added to one of his Mathematical Games 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. Beyond the home domain. No canonical parent is asserted for Ulam's packing conjecture.

Neighborhood in Abstraction Space

Ulam's packing conjecture sits in a sparse region of the domain-specific corpus (66th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Number Theory & Packing Conjectures (5 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08