I was teaching my students about Delaunay Triangulation which is a method for dividing a surface into triangles. This triangulation method is the basis of most computer calculations that require a surface to be approximated as small elements. Here is an example

Example of triangulation

I pointed out that Delaunay's contribution was made in 1934. One student asked why Delaunay would be working on a computer method well before computers were invented. This is a good question that I was wondering if anyone on this site could answer.

The Wickipedia article on Boris Delaunay does mention that he worked on crystal structures so this may be a connection but that is just my speculation. It is interesting that his ideas should be so widely used in areas well outside his interests.

Why did Boris Delaunay work on the very modern topic of triangulation of surfaces when there were no computer methods in his day?

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    $\begingroup$ Why do you say there were no computational methods in 1934? Indeed, Gauss is known for developing computational methods in the 1800s ... "least squares", "Gaussian elimination", and many others. $\endgroup$ Oct 25, 2020 at 21:10
  • $\begingroup$ I have edited. You are correct. I have changed to computer based methods. Is that better? $\endgroup$
    – Hugh
    Oct 25, 2020 at 21:17
  • $\begingroup$ Even "computer methods" is misleading. Before electronic computers, the word "computer" meant a human who does computations. $\endgroup$ Oct 27, 2020 at 15:05
  • $\begingroup$ I am aware of this. What are you suggesting? Should I change to electronic computers? However, how about mechanical computers? $\endgroup$
    – Hugh
    Oct 27, 2020 at 20:51

1 Answer 1


Delaunay (Gallicized version of Russian Delone) did not invent them, they were used long before 1934. Delaunay triangulations, or more generally tesselations, are dual to Voronoi diagrams, the circumcenters of Delaunay triangles are the vertices of the Voronoi diagram. One can infer the motivation from the very title of Delaunay's 1934 paper: Sur la sphere vide. A la memoire de Georges Voronoi.

Dirichlet used such diagrams for regular lattices to study 2D and 3D quadratic forms in 1850 (although they were occasionally used before), and Voronoy introduced a multidimensional generalization in 1908, the year of his death, along with the dual Delaunay tessellations. Delaunay met Voronoy at the University of Warsaw as a teenager, and was strongly influenced. So much so that Mathematical Genealogy Project lists Voronoy as his thesis advisor, even though he defended well after Voronoy’s death. Delaunay's contribution was to study Voronoi diagrams and their duals for irregularly placed points or sites, but even he used them before 1934, e.g. in Neue Darstellung der geometrischen Kristallographie, Z. Kristallograph., 84, 109–149 (1933).

Voronoi diagrams were sporadically used even before Dirichlet, as far back as Kepler and Descartes. Kepler even used Delaunay tessellations to study the shapes of snowflakes and the sphere packing problem. They were also discovered and rediscovered in gold mining, crystallography, metallurgy and meteorology long before computers, e.g. Wigner and Seitz introduced Voronoi diagrams induced by the atoms of a metallic crystal in 1933, without any knowledge of Delaunay's work. See Voronoi diagrams and Delaunay triangulations by Liebling and Pournin for details of applications and a historical sketch.

  • $\begingroup$ Thank you for an amazingly good answer. I have been reading your references. I can recommend the last one. I will have much to tell my students about Delaunay and will have to stop myself giving a history lesson instead of a technical lecture. $\endgroup$
    – Hugh
    Oct 27, 2020 at 10:21
  • $\begingroup$ Delaunay was descended from a French army officer whose surname was "de Launay". The name is therefore not really a "Gallicization" of Делоне. $\endgroup$ Nov 7, 2020 at 14:33

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