# Delaunay simplexes

A **Delaunay simplex** is the dual of the Voronoi diagram. Delaunay simplexes were developed by Борис Николаевич Делоне. In two-dimensions it is more commonly known as *Delaunay triangulation*, and in three-dimensions , as *Delaunay tetrahedralisation*.

A Delaunay triangulation fulfills the *empty circle property* (also called *Delaunay property*): the circumscribing circle of any facet of the triangulation contains no data point in its interior. For a point set with no subset of four co-circular points the Delaunay triangulation is unique. A similar property holds for tetrahedralisation in three dimensions.

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## References

- Математические основы структурного анализа кристаллов (совместно с А.Д.Александровым и Н.Падуровым), Москва, Матем. литература, 1934 г.
- A. V. Anikeenko, M. L. Gavrilova and N. N. Medvedev "A Novel Delaunay Simplex Technique for Detection of Crystalline Nuclei in Dense Packings of Spheres", Lecture Notes in Computer Science
**3480**pp. 816-826 (2005)