Introduction to dessins d'enfants

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A dessin d'enfant is a graph with vertices coloured in two colours

so that no edge has two vertices of the same colour.

They can be drawn on a flat piece of paper, like the one on the left inspired by the Orion constellation

or on a curved surface resembling a doughnut.

These objects were given the name "dessins d'enfants" by Alexander Grothendieck, because they reminded him of something a child would draw while playing (dessin d'enfant is French for "a child's drawing").

"This is surely because of the very familiar, non-technical nature of the objects considered, of which any child’s drawing scrawled on a bit of paper (at least if the drawing is made without lifting the pencil) gives a perfectly explicit example. To such a dessin, we find associated subtle arithmetic invariants, which are completely turned topsy-turvy as soon as we add one more stroke."

"Cela tient sûrement à la nature tellement familière, non technique, des objets considérés, dont tout dessin d’enfant griffonné sur un bout de papier (pour peu que le graphisme soit d'un seul tenant) donne un exemple parfaitement explicite. A un tel dessin se trouvent associés des invariants arithmétiques subtils, qui seront chamboulés complètement dès qu'on y rajoute un trait de plus."

Esquisse d'un Programme
Alexander Grothendieck (1928 - 2014)

More soon to follow!