Packing density¶
A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing.
Core Idea¶
Packing density is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself.
Scope of Application¶
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In compact spaces. If K1,\dots, Kn are measurable subsets of a compact measure space X.
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In compact spaces. and their interiors pairwise do not intersect, then the collection [Ki] is a packing in X and its packing density is.
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In Euclidean space. If the space being packed is infinite in measure, such as Euclidean space, it is customary to define the density as the limit of densities exhibited in balls of larger and.
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In Euclidean space. If Bt is the ball of radius t centered at the origin, then the density of a packing [Ki:i\in\N] is.
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In Euclidean space. \eta = \lim{t\to\infty}\frac{\sum{i=1}^{\infty}\mu(Ki\cap Bt)}{\mu(Bt)}.
Clarity¶
A clear use of Packing density names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing.
Manages Complexity¶
Packing density compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—the ball may also be replaced by dilations of some other convex body, but in general the resulting densities are not equal.—and the practical consequence—if K1,\dots, Kn are measurable subsets of a compact measure space X.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing.
- Check operation and conditions. The optimal packing density or packing constant associated with a supply collection is the supremum of upper densities obtained by packings that are subcollections of the supply collection.
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Packing density transfers literally when a new case preserves the same carrier type, relation, and recognition test. If K1,\dots, Kn are measurable subsets of a compact measure space X. and their interiors pairwise do not intersect, then the collection [Ki] is a packing in X and its packing density is. Beyond the home domain. No canonical parent is asserted for Packing density. 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.
Relationships to Other Abstractions¶
Current abstraction Packing density Domain-specific
Parents (1) — more general patterns this builds on
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Packing density is a kind of Ratio Prime
Packing density is the ratio of occupied measure to total measure in a containing space.
Hierarchy path (1) — routes to 1 parentless root
- Packing density → Ratio → Comparison → Self Checking
Neighborhood in Abstraction Space¶
Packing density sits in a moderately populated region (59th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Combinatorial Optimization & Discrete Structures (31 abstractions)
Nearest neighbors
- Strip packing problem — 0.89
- Filling radius — 0.85
- Stokes's law — 0.84
- Surface-area-to-volume ratio — 0.84
- Characterization (mathematics) — 0.84
Computed from structural-signature embeddings · 2026-10-08