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Indeed, Voronoi tessellations work as a geometrical tool to understand the physical constraints that drive the organization of biological tissues. [22] In hydrology, Voronoi diagrams are used to calculate the rainfall of an area, based on a series of point measurements. In this usage, they are generally referred to as Thiessen polygons.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics , tessellation can be generalized to higher dimensions and a variety of geometries.
A Pythagorean tiling Street Musicians at the Door, Jacob Ochtervelt, 1665.As observed by Nelsen [1] the floor tiles in this painting are set in the Pythagorean tiling. A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t {3,6} (as a truncated triangular tiling).
A circular Dirichlet tessellation with randomly assigned weights. In mathematics, a weighted Voronoi diagram in n dimensions is a generalization of a Voronoi diagram.The Voronoi cells in a weighted Voronoi diagram are defined in terms of a distance function.
The Schläfli symbol can represent a finite convex polyhedron, an infinite tessellation of Euclidean space, or an infinite tessellation of hyperbolic space, depending on the angle defect of the construction. A positive angle defect allows the vertex figure to fold into a higher dimension and loops back into itself as a polytope. A zero angle ...
Cubic honeycomb. In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
The basic equation is: = = ()where () is the estimate at , are the weights and () are the known data at ().The weights, , are calculated by finding how much of each of the surrounding areas is "stolen" when inserting into the tessellation.