Pyjama problem¶

In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are.

Version: v1 · 2026-09-28 · History
Domain-specific #: 11584
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Tiling Theory → Mathematics

Core Idea¶

Pyjama problem is treated here as the recurring tiling theory identity summarized by this source-grounded definition: In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are.

In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are. The problem was posed in 2006 by Alex Iosevich, Mihail Kolountzakis, and Máté Matolcsi. It was answered in the affirmative by Freddie Manners in 2015, using an analogy with Furstenberg’s ×2, ×3 Theorem.

about the origin are sufficient to cover \mathbb{C} , hence obtaining an explicit upper bound for the pyjama problem. Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon . Noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon).

For Pyjama problem, the abstraction is narrower than the article's general subject matter: a positive case must preserve In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in tiling theory, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are.
Constitutive relation — The problem was posed in 2006 by Alex Iosevich, Mihail Kolountzakis, and Máté Matolcsi.
Operating condition — It was answered in the affirmative by Freddie Manners in 2015, using an analogy with Furstenberg’s ×2, ×3 Theorem.
Recognition evidence — Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon .
Admissible variation — Noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon).
Characteristic consequence — about the origin are sufficient to cover \mathbb{C} , hence obtaining an explicit upper bound for the pyjama problem.
Failure boundary — It remains an open problem to obtain lower bounds for the pyjama problem beyond the trivial volume preserving bound of \varepsilon^{-1}/2 .

What It Is Not¶

Not the whole field of tiling theory. The node requires the specific identity stated by In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are.
Not an over-broad reading. Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon .
Not an over-broad reading. Noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon).
Not an over-broad reading. about the origin are sufficient to cover \mathbb{C} , hence obtaining an explicit upper bound for the pyjama problem.
Not automatically Tarski's Plank Problem. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Pyjama problem applies literally inside tiling theory wherever the source-defined carrier and relation can be established. Its documented habitats include:

Quantitative bounds. Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon .
Quantitative bounds. Noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon).
Quantitative bounds. about the origin are sufficient to cover \mathbb{C} , hence obtaining an explicit upper bound for the pyjama problem.
Quantitative bounds. It remains an open problem to obtain lower bounds for the pyjama problem beyond the trivial volume preserving bound of \varepsilon^{-1}/2 .
Documented setting. In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are.
Documented setting. The problem was posed in 2006 by Alex Iosevich, Mihail Kolountzakis, and Máté Matolcsi.

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

Clarity¶

A clear use of Pyjama problem names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are. The strongest recognition evidence in the frozen account is: Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon . A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon . so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Pyjama problem compresses multiple tiling theory details into a stable diagnostic relation. The source shows both the central mechanism—the problem was posed in 2006 by Alex Iosevich, Mihail Kolountzakis, and Máté Matolcsi.—and the practical consequence—about the origin are sufficient to cover \mathbb{C} , hence obtaining an explicit upper bound for the pyjama problem. 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 tiling theory entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are.
Check operation and conditions. It was answered in the affirmative by Freddie Manners in 2015, using an analogy with Furstenberg’s ×2, ×3 Theorem.
Demand recognition evidence. Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon .
Test variation. Change an implementation or setting while preserving noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon).
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 Theory.

Knowledge Transfer¶

Within the home domain. Knowledge about Pyjama problem transfers literally when a new case preserves the same carrier type, relation, and recognition test. Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon . Noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon).

Beyond the home domain. No canonical parent is asserted for Pyjama problem. 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¶

Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon . 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 → In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are; recognition evidence → Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon

Applied / In Practice¶

Noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon). 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 → Quantitative bounds; invariant → In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are; boundary → the case exits the class when let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon

Structural Tensions¶

T1 — Stable identity versus admissible variation. Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon . 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. Noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon). 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. about the origin are sufficient to cover \mathbb{C} , hence obtaining an explicit upper bound for the pyjama problem. 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. It remains an open problem to obtain lower bounds for the pyjama problem beyond the trivial volume preserving bound of \varepsilon^{-1}/2 . 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. In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are. 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 Pyjama problem literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. The problem was posed in 2006 by Alex Iosevich, Mihail Kolountzakis, and Máté Matolcsi. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Pyjama problem distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Pyjama problem is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are. Its framed side is the tiling theory 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: It was answered in the affirmative by Freddie Manners in 2015, using an analogy with Furstenberg’s ×2, ×3 Theorem. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. 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. In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are. 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: In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are. The problem was posed in 2006 by Alex Iosevich, Mihail Kolountzakis, and Máté Matolcsi. It further constrains recognition and variation through: It was answered in the affirmative by Freddie Manners in 2015, using an analogy with Furstenberg’s ×2, ×3 Theorem. Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon .

What is domain-bound. tiling theory supplies the operative entities, technical vocabulary, warrants, and exceptions that make Pyjama problem literal. Its documented scope includes the condition that Let E(\varepsilon) := {z \in \mathbb{C}: \mathrm{Re}(z) \in (-\varepsilon, \varepsilon) \pmod{1}} be the pyjama stripe of width 2\varepsilon . Another bounded application condition is that Noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon). 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—Noah Kravitz and James Leng proved that \exp\exp\exp(\varepsilon^{-O(1)}) rotations of E(\varepsilon).—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 Pyjama problem. The reviewed identity is: In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are. 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¶

Pyjama problem sits in a sparse region of the domain-specific corpus (74^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Combinatorial Optimization & Discrete Structures (31 abstractions)

Nearest neighbors

Strip packing problem — 0.85
Filling radius — 0.83
Coons patch — 0.83
Terminal singularity — 0.83
Julia set — 0.83

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

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are?
Tarski's Plank Problem. Relate geometric coverage to directional thickness: any finite family of Euclidean planks covering a convex body must have total plank width at least the body's minimum width. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Birthday Problem. Pairwise collisions in a finite namespace become likely at the square root of its size, not half of it. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Planarity. The graph property of admitting a crossing-free drawing in the plane — pinned by Kuratowski/Wagner to a finite obstruction (no K₅ or K₃,₃) and by Euler's formula to a density bound, which is why a shelf of NP-hard problems turns polynomial on planar graphs. 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 Pyjama problem remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside tiling theory lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Pyjama_problem (revision 1368141379).
Preserved source candidate: https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf

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.