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.

Version: v3 · 2026-09-19 · History
Domain-specific #: 1638
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Discrete Geometry → Mathematics
Aliases: Delaunay set, Delone point set, (r,R)-set

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.^[1]

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.^[2]

The recognition invariant is metric point set + positive separation scale + finite coverage scale + both bounds holding uniformly over the ambient space.

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Not Too Close, No Big Gaps

Imagine sprinkling dots on a huge floor. Two rules: no two dots can be squished too close together, and there can't be any big empty patch without a dot. A set of dots that follows both rules everywhere is a Delone set. The dots don't have to make a neat repeating pattern.

No Crowds, No Big Holes

A Delone set is a set of points spread across a space, following two rules at once. Rule one: points can never be closer than some fixed distance, so they never crowd together. Rule two: there are no big empty holes, so every spot is within some fixed distance of a point. The same two distances must work everywhere in the whole space. The points do not need to form a repeating pattern like a grid; they just have to be spread evenly enough by both rules.

Separated-and-Covering Point Set

A Delone Set is a set of points in a space where distances make sense (a metric space) that satisfies two conditions at once. It is uniformly discrete: there is a radius r > 0 such that every ball of radius r contains at most one point, so points cannot get arbitrarily close. It is relatively dense: there is a radius R such that every ball of radius R contains at least one point, so there are no arbitrarily large holes. The two conditions push in opposite directions, one limiting crowding and one guaranteeing coverage. The points need not be periodic; crystal lattices are Delone sets, but so are many irregular patterns.

A Delone Set Λ in a metric space is a point set that is both uniformly discrete and relatively dense. In the Euclidean convention, there exist constants r > 0 and R < ∞ such that every open ball of radius r contains at most one point of Λ, and every ball of radius R contains at least one point of Λ. Uniform discreteness rules out arbitrarily close pairs, giving a positive separation scale; relative density rules out holes of unbounded size, giving a finite covering scale. Crucially, both bounds must hold uniformly across the entire ambient space, not just on average or in some region. Their conjunction produces point patterns with controlled local crowding and global coverage without requiring periodicity, so lattices are examples but so are aperiodic patterns. The recognition test is: a point set in a metric space, a positive separation scale, a finite coverage scale, and both bounds holding uniformly.

Structural Signature¶

A declared ambient metric space \((X,d)\).
A subset \(\Lambda\subseteq X\) interpreted as points or sites.
A uniform lower separation bound between distinct sites.
Equivalently, a positive packing radius under a fixed convention.
A uniform upper bound on distance from any ambient point to \(\Lambda\).
Equivalently, a finite covering radius.
Constants that apply globally, not merely locally or on average.
Local finiteness as a consequence in proper metric settings.
No requirement of lattice periodicity.
Compatibility with periodic, quasiperiodic, and disordered examples.
A distinction between the Delone property and stronger finite-local-complexity or repetitivity conditions.
Explicit radius and open/closed-ball conventions when numerical constants matter.

What It Is Not¶

A Delone set is not merely dense: topological density permits points arbitrarily close together and requires approximation at every scale. It is not merely discrete: a discrete set may leave holes whose sizes diverge. It is not necessarily a lattice, crystal, Meyer set, repetitive set, or quasicrystal.^[3]

An \(\varepsilon\)-net under the packing-and-covering convention is a specially coupled case in which a common scale controls both requirements. General Delone constants need not coincide.

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.^[4]

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.

Examples¶

Integer lattice. \(\mathbb{Z}^d\) has positive minimum separation and bounded covering radius, hence is Delone and periodic.

Penrose vertices. The vertex set of a Penrose tiling is Delone but aperiodic, showing that Delone does not mean lattice.

Increasing gaps. \(\{n^2:n\in\mathbb{N}\}\subset\mathbb{R}\) is uniformly discrete but not relatively dense because its gaps grow without bound.

Structural Tensions¶

Separation versus coverage.
Local spacing control versus long-range order.
Flexible constants versus comparable normalization.
Ambient-space coverage versus bounded-window evidence.
Periodic examples versus aperiodic generality.
Geometric abstraction versus physical-site interpretation.

Structural–Framed Character¶

Joint lower and upper bounds are structural. Metric balls, packing, covering, point patterns, and quasicrystalline applications are geometric frame.

Structural Core vs. Domain Accent¶

The portable core is a population constrained to be neither too crowded nor too sparse. The constitutive accent is uniform metric separation and coverage by balls.

This entry is a kind of Constraint.

Constraint is the broader abstraction this entry instantiates. Coverage / Reachability, Balance, Boundedness, Locality, Discrete vs. Continuous, and Spatial Indexing are related.

Relationships to Other Abstractions¶

Current abstraction Delone Set Domain-specific

Parents (1) — outgoing typed relations

Delone Set is a kind of Constraint Prime

Constraint is the broader abstraction this entry instantiates.

Hierarchy path (1) — routes to 1 parentless root

Delone Set → is a kind of Constraint

Neighborhood in Abstraction Space¶

Delone Set sits in a sparse region of the domain-specific corpus (70^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Point-Set Topology & Measure Structures (14 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With¶

Dense set in topology.
Arbitrary discrete subset.
Lattice or ideal crystal.
Meyer set.
Epsilon-covering alone.
Finite local complexity or repetitivity.

References¶

[1] Jeffrey C. Lagarias and Peter A. B. Pleasants, “Repetitive Delone Sets and Quasicrystals,” Ergodic Theory and Dynamical Systems 23 (2003): 831–867, arXiv:math/9909033. registry ↩ Show verification details

[2] Bielefeld Tilings Encyclopedia, “Delone Set,” https://tilings.math.uni-bielefeld.de/glossary/delone-set/. registry ↩

[3] Jeffrey C. Lagarias, “Mathematical Quasicrystals and the Problem of Diffraction,” in Directions in Mathematical Quasicrystals (AMS, 2000).not authoritative registry ↩

[4] Yasushi Nagai and Daniel Lenz, “A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation,” Discrete & Computational Geometry 62 (2019): 603–632, doi:10.1007/s00454-019-00081-2. registry ↩ Show verification details