mathematics

Ulam's packing conjecture

Ulam's packing conjecture is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ulam's packing conjecture rather than just read about it. In short: 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.

Key takeaways

Ulam's packing conjecture belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ulam's packing conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ulam's packing conjecture from memory before moving on to harder problems.

Reference excerpt

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.

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. 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, 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 Kepler conjecture remain open.

References

Illustrations

Worked examples

Example 1 — a first encounter with Ulam's packing conjecture

Start with the simplest possible case. Write down what Ulam's packing conjecture claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ulam's packing conjecture before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ulam's packing conjecture ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ulam's packing conjecture

In research: Ulam's packing conjecture appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ulam's packing conjecture in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ulam's packing conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conjectures, Packing problems, Spheres, so understanding it makes those chapters shorter.
In everyday life: Look for Ulam's packing conjecture outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Ulam's packing conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ulam's packing conjecture means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ulam's packing conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ulam's packing conjecture in simple terms?

Why does Ulam's packing conjecture matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ulam's packing conjecture?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ulam's packing conjecture.