Disk-Covering Problem¶

One of the covering disks is placed central and the remaining five in a symmetrical way around it.

Version: v1 · 2026-09-28 · History
Domain-specific #: 9010
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Discrete Geometry, Covering Problems → Mathematics

Core Idea¶

Disk-Covering Problem is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: One of the covering disks is placed central and the remaining five in a symmetrical way around it.

The disk covering problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk. Dually, for a given radius ε, one wishes to find the smallest integer n such that n disks of radius ε can cover the unit disk. The best solutions known to date are as follows.

One of the covering disks is placed central and the remaining five in a symmetrical way around it. The corresponding angles θ are written in the "Symmetry" column in the above table. While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively.

For Disk-Covering Problem, the abstraction is narrower than the article's general subject matter: a positive case must preserve One of the covering disks is placed central and the remaining five in a symmetrical way around it. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics, logic, and statistics, which is why this identity is domain-specific rather than prime.

How would you explain it like I'm…

The Sticker-Cover Puzzle

Imagine a round cookie that you want to cover completely with smaller round stickers, all the same size. If you get a certain number of stickers, how small can they be and still cover the whole cookie? For some numbers of stickers, the best way people have found is to put one sticker in the middle and the rest in a neat ring around it.

Smallest Circles to Cover a Circle

The disk-covering problem is a math puzzle about circles. You have one circle of size 1, and a set number of smaller circles that are all the same size. The question is: what's the smallest size those circles can be and still cover every bit of the big circle? You can also ask it the other way: if the small circles have a fixed size, what's the fewest you need? For seven to ten circles, the best arrangements known put one circle in the center with the others spaced evenly around it.

Minimal-Radius Disk Covering

The disk-covering problem asks for the smallest radius r(n) such that n equal disks of that radius can be arranged to cover the unit disk completely. The dual version fixes the radius ε and asks for the smallest number of disks of radius ε that can cover the unit disk. Solutions are given as the best known arrangements for each n. One notable pattern places one disk at the center and the rest symmetrically around it: one central disk with five around it is not the best layout for r(6), but one central disk with six, seven, eight, or nine disks around it gives the best known layouts for r(7), r(8), r(9), and r(10). The positions in such arrangements are described by the angles of the surrounding disks.

The disk-covering problem seeks, for each n, the smallest real number r(n) such that n disks of radius r(n) can be placed to cover the unit disk; dually, for a given radius ε, it seeks the minimal number n of radius-ε disks that cover the unit disk. Results are reported as best-known solutions, specified by the disk arrangement and the resulting radius. A recurring structure is a central disk surrounded by a symmetric ring of equal disks, with the ring's angular positions recorded as a symmetry parameter. For six disks, a central disk plus five symmetrically placed disks is not the optimal layout. However, analogous arrangements with six, seven, eight, and nine disks around a central disk of the same radius provide the best known layouts for r(7), r(8), r(9), and r(10). The problem is a specific covering question about equal disks and the unit disk, not covering problems in general.

Structural Signature¶

Sig role-phrases:

Defining carrier — The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6.
Constitutive relation — One of the covering disks is placed central and the remaining five in a symmetrical way around it.
Operating condition — The corresponding angles θ are written in the "Symmetry" column in the above table.
Recognition evidence — While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively.
Admissible variation — The disk covering problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk.
Characteristic consequence — Dually, for a given radius ε, one wishes to find the smallest integer n such that n disks of radius ε can cover the unit disk.
Failure boundary — The best solutions known to date are as follows.

What It Is Not¶

Not the whole field of mathematics, logic, and statistics. The node requires the specific identity stated by One of the covering disks is placed central and the remaining five in a symmetrical way around it.
Not an over-broad reading. While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively.
Not an over-broad reading. The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6.
Not an over-broad reading. One of the covering disks is placed central and the remaining five in a symmetrical way around it.
Not automatically Smallest-Circle Problem. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Disk-Covering Problem applies literally inside mathematics, logic, and statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Method. The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6.
Method. One of the covering disks is placed central and the remaining five in a symmetrical way around it.
Method. The corresponding angles θ are written in the "Symmetry" column in the above table.
Method. While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively.
Documented setting. The disk covering problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk.
Documented setting. Dually, for a given radius ε, one wishes to find the smallest integer n such that n disks of radius ε can cover the unit disk.

Outside mathematics, logic, and statistics, 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 Disk-Covering Problem names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is One of the covering disks is placed central and the remaining five in a symmetrical way around it. The strongest recognition evidence in the frozen account is: While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Disk-Covering Problem compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—one of the covering disks is placed central and the remaining five in a symmetrical way around it.—and the practical consequence—dually, for a given radius ε, one wishes to find the smallest integer n such that n disks of radius ε can cover the unit disk. 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 mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: One of the covering disks is placed central and the remaining five in a symmetrical way around it.
Check operation and conditions. The corresponding angles θ are written in the "Symmetry" column in the above table.
Demand recognition evidence. While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively.
Test variation. Change an implementation or setting while preserving the disk covering problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk.
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 Disk-Covering Problem transfers literally when a new case preserves the same carrier type, relation, and recognition test. The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. One of the covering disks is placed central and the remaining five in a symmetrical way around it.

Beyond the home domain. No canonical parent is asserted for Disk-Covering 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¶

The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. 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 → One of the covering disks is placed central and the remaining five in a symmetrical way around it; recognition evidence → While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively

Applied / In Practice¶

One of the covering disks is placed central and the remaining five in a symmetrical way around it. 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 → Method; invariant → One of the covering disks is placed central and the remaining five in a symmetrical way around it; boundary → the case exits the class when while this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively

Structural Tensions¶

T1 — Stable identity versus admissible variation. While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively. 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. The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. 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. One of the covering disks is placed central and the remaining five in a symmetrical way around it. 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. The corresponding angles θ are written in the "Symmetry" column in the above table. 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. The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. 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 Disk-Covering Problem literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. One of the covering disks is placed central and the remaining five in a symmetrical way around it. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Disk-Covering Problem distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Disk-Covering Problem is structural-leaning. Its structural side is the repeatable organization summarized by One of the covering disks is placed central and the remaining five in a symmetrical way around it. Its framed side is the mathematics, logic, and statistics 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: The corresponding angles θ are written in the "Symmetry" column in the above table. 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. One of the covering disks is placed central and the remaining five in a symmetrical way around it. 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: The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. One of the covering disks is placed central and the remaining five in a symmetrical way around it. It further constrains recognition and variation through: The corresponding angles θ are written in the "Symmetry" column in the above table. While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively.

What is domain-bound. mathematics, logic, and statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Disk-Covering Problem literal. Its documented scope includes the condition that The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. Another bounded application condition is that One of the covering disks is placed central and the remaining five in a symmetrical way around it. 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—The disk covering problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk.—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 Disk-Covering Problem. The reviewed identity is: One of the covering disks is placed central and the remaining five in a symmetrical way around it. 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¶

Disk-Covering Problem sits in a moderately populated region (57^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Combinatorial Optimization & Discrete Structures (31 abstractions)

Nearest neighbors

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 One of the covering disks is placed central and the remaining five in a symmetrical way around it?
Smallest-Circle Problem. The smallest-circle problem asks for the unique minimum-radius Euclidean disk containing a finite set of planar points, equivalently minimizing the maximum point-to-center distance in the two-dimensional Euclidean 1-center case. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Covering number. The minimum number of radius-r balls required to cover a specified subset of a metric or pseudometric space. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Unit disk graph. The intersection graph of equal-radius disks in the Euclidean plane, equivalently a graph connecting points whose pairwise distance is at most a fixed threshold after scaling. 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 Disk-Covering Problem remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics, logic, and statistics 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/Disk_covering_problem (revision 1270913042).
Preserved source candidate: https://erich-friedman.github.io/packing/circovcir/

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.