Voidsgaps

Voidsgaps is a term used in the study of spatial distributions and porous structures to describe a recurring pattern in which regions of a domain are alternately empty (voids) and sparsely populated (gaps). The concept is used across disciplines such as materials science, cosmology, and data geometry to quantify irregularity and to compare different systems by their arrangement of voids and gaps.

Definition: In a metric space with a finite point set P and a scale parameter r, an r-void is a maximal connected region that lies at distance greater than r from every point in P. A gap is a connected region where the local density of points falls below a threshold but that is not entirely void. The void-gap pattern then refers to the sequence or map of all such voids and gaps as r varies.

Computation and properties: The pattern is scale- and threshold-dependent and is not unique to a single dataset. It can be analyzed with tools from computational geometry and topology, using Voronoi diagrams, Delaunay triangulations, alpha shapes, density-based clustering, and persistent homology to identify and measure voids and gaps, including metrics such as void fraction, average gap depth, and inter-void distances.

Applications: Understanding pore structure in materials, characterizing large-scale structure in cosmology, evaluating coverage in sensor networks, and informing planning problems where spatial gaps affect performance.

See also: void, gap, persistent homology, Voronoi diagram, alpha shapes.