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. 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 about sphere packing. 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. Nevertheless, there is an infinite space of possible shapes that have not been ruled out.

For Ulam's packing conjecture, the abstraction is narrower than the article's general subject matter: a positive case must preserve 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. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in discrete geometry, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — 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.
Constitutive relation — In dimensions above four (excluding 8 and 24), the situation is complicated by the fact that the analogs of the Kepler conjecture remain open.
Operating condition — 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.
Recognition evidence — 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.
Admissible variation — Nevertheless, there is an infinite space of possible shapes that have not been ruled out.
Characteristic consequence — Yoav Kallus has shown that at least among point-symmetric bodies, the ball constitutes a local maximum of the fraction of empty space forced.
Failure boundary — That is, any point-symmetric solid that does not deviate too much from a ball can be packed with greater efficiency than can balls.

What It Is Not¶

Not the whole field of discrete geometry. The node requires the specific identity stated by 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.
Not an over-broad reading. Nevertheless, there is an infinite space of possible shapes that have not been ruled out.
Not an over-broad reading. That is, any point-symmetric solid that does not deviate too much from a ball can be packed with greater efficiency than can balls.
Not an over-broad reading. However, the regular octagon and smoothed octagon give counter-examples.
Not automatically Sphere packing. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Ulam's packing conjecture applies literally inside discrete geometry wherever the source-defined carrier and relation can be established. Its documented habitats include:

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 1972.
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.
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.

Outside discrete geometry, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

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. The strongest recognition evidence in the frozen account is: 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. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Nevertheless, there is an infinite space of possible shapes that have not been ruled out. so that a reader can reproduce the classification rather than infer it from topical resemblance.

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 forced. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the discrete geometry entities to which the claim applies.
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.
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 been ruled out as counterexamples of Ulam's conjecture.
Demand recognition evidence. 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.
Test variation. Change an implementation or setting while preserving nevertheless, there is an infinite space of possible shapes that have not been ruled out.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

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. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

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. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → 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; recognition evidence → 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

Applied / In Practice¶

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. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Supporting arguments; invariant → 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; boundary → the case exits the class when nevertheless, there is an infinite space of possible shapes that have not been ruled out

Structural Tensions¶

T1 — Stable identity versus admissible variation. Nevertheless, there is an infinite space of possible shapes that have not been ruled out. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. That is, any point-symmetric solid that does not deviate too much from a ball can be packed with greater efficiency than can balls. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. However, the regular octagon and smoothed octagon give counter-examples. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. 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. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. 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. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Ulam's packing conjecture literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. In dimensions above four (excluding 8 and 24), the situation is complicated by the fact that the analogs of the Kepler conjecture remain open. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Ulam's packing conjecture distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Ulam's packing conjecture is mixed or framed-leaning. Its structural side is the repeatable organization summarized by 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. Its framed side is the discrete geometry vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: 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. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. 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 stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: 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. In dimensions above four (excluding 8 and 24), the situation is complicated by the fact that the analogs of the Kepler conjecture remain open. It further constrains recognition and variation through: 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. 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.

What is domain-bound. discrete geometry supplies the operative entities, technical vocabulary, warrants, and exceptions that make Ulam's packing conjecture literal. Its documented scope includes the condition that 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. Another bounded application condition is that 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. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Nevertheless, there is an infinite space of possible shapes that have not been ruled out.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Ulam's packing conjecture. The reviewed identity 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. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Ulam's packing conjecture sits in a sparse region of the domain-specific corpus (66^th 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

Kemnitz's Conjecture — 0.86
Wang tile — 0.85
Simplicial sphere — 0.84
Prototile — 0.84
Formal theorem — 0.84

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

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish 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?
Sphere packing. Arrange nonoverlapping equal-radius balls in a specified ambient space to maximize a declared finite or asymptotic density under explicit boundary, periodicity, and congruence conventions. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Packing dimension. A fractal dimension defined from the critical exponent of disjoint small-ball packings after a countable-cover regularization. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Tetrahedron packing. The geometric optimization problem of arranging congruent regular tetrahedra without overlapping so as to maximize the fraction of three-dimensional space they occupy. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Ulam's packing conjecture remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside discrete geometry lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Ulam%27s_packing_conjecture (revision 1314754857).
Preserved source candidate: https://archive.org/details/newmathematicald00mgar
Preserved source candidate: https://archive.org/details/newmathematicald00mgar/page/n250

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.