The basic structures of Voronoi and generalized Voronoi polygons

Miles, R. E.; Maillardet, Robert

doi:10.2307/3213553

Cited by 36 publications

(7 citation statements)

References 6 publications

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“…We have chosen Voronoi tessellations to explore the emergence of diverse organizations in packed tissues. It has been described that a random distribution of seeds in the Euclidean plane produces a Poisson–Voronoi diagram with a fixed polygon distribution that is independent of the number of starting seeds (Miles & Maillardet, ). We have reproduced these results (Fig F and G) and also obtained histograms enriched in hexagons (29.5%), pentagons (25.9%) and heptagons (19.9%), among other polygons (Fig E Diagram 1 and Table EV1).…”

Section: Resultsmentioning

confidence: 99%

Fundamental physical cellular constraints drive self‐organization of tissues

Sánchez‐Gutiérrez

Tozluoğlu

Barry³

et al. 2015

The EMBO Journal

116

157

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Morphogenesis is driven by small cell shape changes that modulate tissue organization. Apical surfaces of proliferating epithelial sheets have been particularly well studied. Currently, it is accepted that a stereotyped distribution of cellular polygons is conserved in proliferating tissues among metazoans. In this work, we challenge these previous findings showing that diverse natural packed tissues have very different polygon distributions. We use Voronoi tessellations as a mathematical framework that predicts this diversity. We demonstrate that Voronoi tessellations and the very different tissues analysed share an overriding restriction: the frequency of polygon types correlates with the distribution of cell areas. By altering the balance of tensions and pressures within the packed tissues using disease, genetic or computer model perturbations, we show that as long as packed cells present a balance of forces within tissue, they will be under a physical constraint that limits its organization. Our discoveries establish a new framework to understand tissue architecture in development and disease.

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Section: Resultsmentioning

confidence: 99%

Fundamental physical cellular constraints drive self‐organization of tissues

Sánchez‐Gutiérrez

Tozluoğlu

Barry³

et al. 2015

The EMBO Journal

116

157

View full text Add to dashboard Cite

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“…At a basic level, a Voronoi diagram partitions the space around a set of focal points, such that each point is contained in a single polygonal zone. The boundaries of each Voronoi polygon are defined such that all areas within a given focal point's polygon are closer to that focal point than to any other focal point (Aurenhammer, 1991;Miles & Maillardet, 1982;Okabe, Boots, Sugihara, & Chiu, 2000). Applied to the context of schools, a Voronoi diagram for each school district is constructed around all of the schools in the district that serve students of the same grade level.…”

Section: Voronoi Polygonal Attendance Zonesmentioning

confidence: 99%

The Gerrymandering of School Attendance Zones and the Segregation of Public Schools

Richards

2014

American Educational Research Journal

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In this study, I employ geospatial techniques to assess the impact of school attendance zone ''gerrymandering'' on the racial/ethnic segregation of schools, using a large national sample of 15,290 attendance zones in 663 districts. I estimate the effect of gerrymandering on school diversity and school district segregation by comparing the racial/ethnic characteristics of existing attendance zones to those of counterfactual zones expected in the absence of gerrymandering. Results indicate that the gerrymandering of attendance zones generally exacerbates segregation, although it has a weaker effect on the segregation of Whites from Blacks and Hispanics. Gerrymandering is particularly segregative in districts experiencing rapid racial/ethnic change. However, gerrymandering is associated with reductions in segregation in a substantial minority of districts, notably those under desegregation orders.

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“…A mathematically rigorous and self-contained presentation of the basic material (mean-value relationships, PVT's, stochastic-and integral-geometric tools) on random VT's can be found in Møller (1994). For further details and the historical background the reader is also referred to the monographs of Okabe et al (2000) and Stoyan et al (1995) (in particular Chapter 10 and references therein) and the papers by Miles (1970), Miles and Maillardet (1982), Møller (1989) and others. Because there are only a few explicit formulae of distributional characteristics of PVT's (see e.g.…”

Section: Basic Definitions and Preliminary Resultsmentioning

confidence: 99%

Second‐order properties of the point process of nodes in a stationary Voronoi tessellation

Heinrich

Muche

2008

Mathematische Nachrichten

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Key words Voronoi tessellation, stationary Poisson process, point process of nodes, second moment measure, pair correlation function, asymptotic variance, central limit theorem, vertices of the typical cell MSC (2000) Primary: 60D05, 60G55; Secondary: 62F12, 60G60In this paper we derive representation formulae for the second factorial moment measure of the point process of nodes and the second moment of the number of vertices of the typical cell associated with a stationary normal Voronoi tessellation in R d . In case the Voronoi tessellation is generated by a stationary Poisson process with intensity λ > 0 the corresponding pair correlation function g V,λ (r) can be expressed by a weighted sum of d+2 (numerically tractable) multiple parameter integrals. The asymptotic variance of the number of nodes in an increasing cubic domain as well as the second moment of the number of vertices of the typical Poisson Voronoi cell are calculated exactly by means of these parameter integrals. The existence of a (d − 1)st-order pole of g V,λ (r) at r = 0 is proved and the exact value of limr→0 r d−1 g V,λ (r) is determined. In the particular cases d = 2 and d = 3 the graph of gV,1(r) including its local extreme points, the points of level 1 of gV,1(r) and other characteristics are computed by numerical integration. Furthermore, an asymptotically exact confidence interval for the intensity of nodes is obtained.A tessellation of the d-dimensional Euclidean space R d is a subdivision of this space into countably many compact sets {C i , i ∈ N} (called cells or crystals), where two such cells have no common interior points and the number of cells intersecting any bounded subset of R d is finite. Usually the cells are assumed to be convex so that each C i becomes a convex d-polytope. In this case the boundary ∂C i consists of s-polytopes with s ∈ {0, 1, . . . , d−1} called s-faces in what follows. As usual 0-faces, 1-faces, and (d − 1)-faces are called vertices, edges, and facets, respectively. The union of all vertices forms the countable set of nodes of the tessellation.Many real-life tessellations are random, see e.g. Stoyan et al. (1995), Okabe et al. (2000. There are various stochastic-geometric mechanisms to generate a random tessellation {C i , i ∈ N} which can be defined over a hypothetical probability space [Ω, A, P] such that the above properties are satisfied P-almost surely. The random tessellation is said to be stationary if its distribution is invariant under translation of the cells.Throughout the present paper we are concerned with stationary Voronoi (or Dirichlet, or Thiessen) tessellations (briefly VT's) generated by stationary point processes. It should be mentioned that some authors use the term VT merely relative to a homogeneous Poisson point process (henceforth such a VT will be called Poisson Voronoi tessellation (briefly PVT)). A mathematically rigorous and self-contained presentation of the basic material (mean-value relationships, PVT's, stochastic-and integral-geometric tools) on random VT's can be found...

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The basic structures of Voronoi and generalized Voronoi polygons

Cited by 36 publications

References 6 publications

Fundamental physical cellular constraints drive self‐organization of tissues

Fundamental physical cellular constraints drive self‐organization of tissues

The Gerrymandering of School Attendance Zones and the Segregation of Public Schools

Second‐order properties of the point process of nodes in a stationary Voronoi tessellation

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