tessellation: one construct, many systems

In biology, tessellation is the space-filling arrangement of cells or units that partition a surface or volume without gaps or overlaps, as in the near-hexagonal packing of cells in a tissue or a honeycomb. D'Arcy Thompson's On Growth and Form showed how physical constraints such as surface tension drive living matter toward these regular space-filling patterns. The referent is a gap-free partition of biological space into cells, the same construct that geometry studies as plane tilings and that spatial analysis builds as Voronoi diagrams.[1]

Unit

The gap-free partition of a tissue or surface into packed cells

Recovery means

Not applicable

Measured by

Quantified by cell shape, number of neighbours, and how closely packing approaches a regular space-filling pattern

In geometry, a tessellation (tiling) is a covering of the plane by shapes that meet edge to edge with no gaps and no overlaps, classified by their symmetries and the tiles they use. Grünbaum and Shephard's Tilings and Patterns is the comprehensive treatment of how such tilings are built, classified, and constrained. The referent is a mathematical partition of the plane into tiles, the same space-filling construct realised physically in packed biological cells and generalised in spatial statistics as the Voronoi tessellation.[2]

Unit

A gap-free, non-overlapping covering of the plane by tiles

Recovery means

Not applicable

Measured by

Classified by symmetry group, tile shapes, and the vertex configurations of the tiling

In spatial analysis and computational geometry, a tessellation is a partition of space into cells, most often the Voronoi tessellation, which assigns to each of a set of points the region closer to it than to any other point. Okabe, Boots, Sugihara, and Chiu's Spatial Tessellations develops Voronoi and related partitions and their uses across the sciences. It is the same gap-free space-partition construct seen in packed cells and plane tilings, here driven by a set of generating points and measured by the resulting cell geometry.[3]

Unit

The partition of space into cells defined by proximity to generating points

Recovery means

Not applicable

Measured by

Computed from generator points as Voronoi cells, with cell area, neighbours, and boundaries analysed

In neuroscience, tiling is the way neurons of a single type arrange their dendrites or receptive fields so that together they cover the sensory surface completely but with little overlap, like floor tiles that meet edge to edge. Wässle, Peichl, and Boycott's 1981 Dendritic territories of cat retinal ganglion cells showed that each class of retinal ganglion cell tiles the retina, its members' dendritic fields filling the surface without redundancy. The referent is complete, non-redundant coverage of a space by a population, measured by coverage factor, and it is the same construct mathematicians study as the tiling of a plane.[4]

Unit

How completely and non-redundantly a cell population covers its surface

Recovery means

Not applicable

Measured by

Quantified by the coverage factor: dendritic-field area times cell density