Delone Set¶
A metric-space point set with both a positive uniform-separation bound and a finite covering-radius bound, so it is nowhere arbitrarily crowded and nowhere arbitrarily sparse.
Core Idea¶
A Delone set \(\Lambda\) in a metric space is simultaneously uniformly discrete and relatively dense. In Euclidean convention, there are constants \(r>0\) and \(R<\infty\) such that every ball of radius \(r\) contains at most one point of \(\Lambda\), while every ball of radius \(R\) contains at least one.
The two bounds work in opposite directions. Uniform discreteness prevents arbitrarily close pairs; relative density prevents holes of unbounded size. Their conjunction creates a point pattern with controlled local crowding and global coverage without requiring periodicity.
Scope of Application¶
Delone sets model atomic sites in crystals and quasicrystals, vertices of tilings, sampling sites, geometric codes, and controlled point clouds. Their packing and covering radii mediate error correction, facility coverage, meshing, and approximation. Adding finite local complexity, repetitivity, diffraction, or difference-set restrictions produces important subclasses, but those conditions are not part of the base identity.
Clarity¶
State the ambient space, metric, set, separation convention, and covering convention. Authors variously use \(r\) for minimum distance or for packing radius, introducing a factor of two. A finite point set is not relatively dense in an unbounded ambient space simply because it covers a bounded sample region.
Manages Complexity¶
Two scalars summarize a point pattern's most basic geometric adequacy: the closest allowed crowding and the farthest allowed gap. This supports compactness arguments, local patch enumeration, algorithmic bounds, and comparisons across patterns without assuming coordinates repeat.
Abstract Reasoning¶
- Fix the ambient metric space and ball convention.
- Compute or bound the infimum of pairwise distances.
- Verify that a strictly positive uniform separation scale exists.
- Compute or bound \(\sup_{x\in X} d(x,\Lambda)\).
- Verify that this covering radius is finite.
- Record Delone constants under the chosen convention.
- Test any stronger claim—periodicity, repetitivity, finite local complexity, Meyer property—separately.
- Preserve the ambient space when comparing or transforming point patterns.
Knowledge Transfer¶
The portable pattern is a dual-sided spacing constraint: prohibit both excessive concentration and excessive absence. The proposed immediate parent is Constraint.
Relationships to Other Abstractions¶
Current abstraction Delone Set Domain-specific
Parents (1) — more general patterns this builds on
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Delone Set is a kind of Constraint Prime
Constraint is the proposed immediate parent.
Hierarchy path (1) — routes to 1 parentless root
- Delone Set → Constraint
Neighborhood in Abstraction Space¶
Delone Set sits in a sparse region of the domain-specific corpus (74th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Metric Geometry & Approximation (13 abstractions)
Nearest neighbors
- Sphere packing — 0.90
- Equilateral Dimension — 0.86
- Reach (Mathematics) — 0.82
- Metric projection — 0.82
- Tarski's Plank Problem — 0.82
Computed from structural-signature embeddings · 2026-09-08