## 6069 - From the Voronoi diagrams to the Delaunay triangulation

### N. Lygeros

Translation: Paola Vagioni

The Voronoi diagrams do not only solve problems in geography and topology. When Georgy Voronoi (1868-1908) invented them, who was a student of Andrey Markov (1856-1922) but also the teacher of Delaunay (1890-1980) and Sierpiński (1882-1969), he created an entire mathematical framework via computational geometry. The idea of the Voronoi diagrams is the creation of new points that differ from the initial facts and are not presented in a simple way to the solver. They create a new structure which is invisible to the non-expert who examines the initial elements. This notion is most important in mathematics and not only there. On the cognitive level, it releases the thought and makes it function non-conventionally for solving a seemingly static problem. The contibution of the Voronoi diagrams is the introduction of a dynamic framework, which facilitates not only the solution but also the invention of a new strategy. Moreover, the Voronoi diagrams are directly related to the triangulation which was invented by Delaunay in 1934. Specifically, the Voronoi diagrams and the Delaunay triangulation are dually combined on a general case. The interesting part is that in practice, the Voronoi diagrams are not related with circles but with line segments. While the Delaunay triangulation is defined exclusively via circles. The centres of the triangles of triangulation which do not include any initial point, if we combine them we will find a Voronoi diagram. For this reason we say that they operate in a dual manner. This means that in the framework of the application of the Voronoi diagrams, there is the possibility to exploit also the Delaunay triangulation in order to use its triangles too, which can include or not the Voronoi diagrams. From the Voronoi diagrams to the Delanay triangulation there is a mental schema with multiple applications also beyond the space of pure mathematics and especially in the space of strategy via topostrategy. This mental schema operates as a manifold for the cognitive approach of mental strategy.