<jupyter_start><jupyter_text>Introduction to Voronoi Tessellation Method  Introduction  **Figure 1.** <NAME>, Russian mathematicianAccording to the online Merriam-Webster, the word `tessellation` has two meanings: (a) mosaic; and (b) "a covering of an infinite geometric plane without gaps or overlaps by congruent plane figures of one type or a few types". The word is derived the the Latin, `tessella`, a small cubical piece of clay, stone or glass used to make mosaics (from Wikipedia). The Russian mathematician, from whose name the term `Voronoi` was coined, is <NAME>, shown in the picture.    Some notes from Wikipedia: **Figure 2.** A euclidean distance Voronoi diagram (from Wikipedia)>  > <NAME> (28 April 1868 – 20 November 1908) was a Russian mathematician noted for defining the Voronoi diagram.[2]>  > Among his students was <NAME> (Ph.D. at Jagiellonian University in 1906). Although he was not formally the doctoral advisor of <NAME> (Ph.D. at Kiev University), his influence on the latter earns him the right to be considered so.[3]  Uses and applications, short history From [this webpage](https://www.osti.gov/servlets/purl/1090872) from the Office of Scientific and Technology Information, US Department of Energy (https://www.osti.gov) and from [1]: Dr. <NAME> and Voronoi tessellations> The first documented application of Voronoi tessellations appeared in the classictreatise of Snow (Snow, 1855) on the 1854 cholera epidemic in London in which hedemonstrated that proximity to a particular well was strongly correlated to deathsdue to the disease.  Short history> Voronoi tessellation has a long history, probably because Voronoi-like arrangements often appear in nature. Voronoi-like tessellations appeared in 1644 in thework of Decartes on the distribution of matter in the cosmic region near our sun.The first systematic treatment of what we now call Voronoi tessellations was givenby Dirichlet (Dirichlet, 1850) in his study of two- and three-dimensional quadraticforms, i.e., homogeneous, multivariate polynomials of degree two; hence, Voronoiregions are often referred to as Dirichlet cells. Voronoi (Voronoi, 1907) generalizedthe work of Dirichlet to arbitrary dimensions, again using what are now referred toas Voronoi tessellations or diagrams. A method with many names...> It is not surprising, due to their ubiquity and usefulness, that throughout the 20thcentury, **Voronoi tessellations** were rediscovered many times. As a result, Voronoiregions have been called by many different names. **Thiessen polygons** refer to thework of Theissen on developing more accurate estimates for the average rainfallin a region. **Area of influence polygons** was a term coined in connection with theprocessing of data about ore distributions obtained from boreholes. **Wigner-Seitz regions**, domain of an atom, and **Meijering cells** were terms that arose from work oncrystal lattices and the Voronoi cell of the reciprocal crystal lattice is referred to asthe **Brillouin zone** (Kittel, 2004; Ziman, 1979). In the study of codes by, e.g., Shannon, Voronoi cells are called **maximum likelihood regions** (Weaver and Shannon,1963). The field of ecology gave rise to two more alternate labels: **area potentiallyavailable** and **plant polygons** for a Voronoi region associated with a particular treeor plant. **Capillary domains** refers to Voronoi regions in a tissue based on the centersof capillaries.<jupyter_code>show_youtube('https://www.youtube.com/embed/65XBl7y9mSM', 800,500)<jupyter_output><empty_output>
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