The Closest Possible

By on 03/06/2017
Euclidean Voronoi diagram by Balu Ertl

A new contribution from the Italian journal XlaTangente. Initially thought as the Italian edition of the French magazine “Tangente. L’aventure mathématique“, in the last few years the paper edition evolved in a website,, whose goal is to continue to offer “chewable mathematics”, thanks to the contributions of young mathematicians, increasingly aware of the need to maintain a dialogue with society. The website is addressed specifically to students of secondary schools and their teachers.
This contribution was first published in XlaTangente no. 39, June 2013. 

In winter time, you will often see cars with the windshield covered with a thin layer of ice, which is caused by the combined effect of humidity and cold.

This layer is due to the creation (in random times and, in general, even in random positions) of very small, even infinitesimal, ice crystals, covering the whole windshield; in other words, the windshield is “tessellated” by ice crystals which appear and grow during the night.

Well, for crystallographers (and also for mathematicians, who, speaking quite frankly, can sometimes show unusual curiosity) it is interesting to study the final geometric structure of the crystal mosaic on the windshield and, in particular, of the surfaces, said crystal interfaces, along which two crystals meet.  In other words, they want to propose a mathematical model that could describe the final mosaic made up of crystals formed over time.

Initially, as with all models that describe a real phenomenon, it is necessary to make some simplifications or, better, some assumptions; in particular, suppose that:

  • the windscreen is perfectly flat;
  • the process started simultaneously at certain points (that are called generators) where crystals of extremely small size have appeared;
  • generators stay fixed throughout the growth process (i.e., the crystals cannot move on the windshield);
  • all crystals grow at the same speed;
  • each crystal grows with constant speed along all the “free” (available) directions (that is, in each moment, crystals are circles);
  • each crystal stops growing along a given direction if the latter is already occupied by another crystal (that is the crystal ceases to be a circle after touching another crystal).

How many hypotheses! In fact, those are reasonable assumptions which allow us to make some predictions on the mosaic that will appear. For example, after a long enough time, the entire windshield will be covered with crystals. In this way, the windshield can be split in crystals that meet along crystal interfaces. This partitioning forms a tessellation that shows a beautiful property:

The points of the mosaic of crystals centered in a generator are closer to that generator than to any other.

Partitionings of this type are called Voronoi diagrams after Russian mathematician Georgy Voronoi (1868-1908).

Describing Voronoi diagrams and their applications in the real world is not trivial: beyond the example of the windshield (for which I am grateful to Professor Vincenzo Capasso), we can quote the one of the astronomer interested in the structure of the universe, or that of the archeologist trying to identify influential zones of different tribes or even that of the city administration which has to plan its dislocation of schools/hospitals/public offices, that of the phone operator that has to decide where to place its repeaters, that of the physiologist who studies how muscle tissue is supplied with oxygen by the capillaries, and so on.

Voronoi diagrams

Given a finite set S of points in the plane, a Voronoi diagram for that set is the partition of the plane that associates to each point p of S, a region, called a cell, so that all points of the cell are closer to the point p than to any other point in S. You can see an Euclidean Voronoi diagram in the image above, by Balu Ertl.


Enea G. Bongiorno,
translated by Daniela Della Volpe

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