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Jérémie Bettinelli

École polytechnique
Laboratoire d'informatique (LIX)
91128 Palaiseau Cedex
FRANCE
E-mail: firstname « . » lastname « at » normalesup « . » org
Office: 2023
Phone: (+33) (0)1 77 57 80 61
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Plane quadrangulations

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Sketchfab folderPicture folder

For many reasons, the quadrangulations constitute an important family of maps. A quadrangulation is simply a map whose faces are all incident to 4 half-edges (basically, they are deformable squares). It is called a plane (or sometimes planar) quadrangulation when it is a quadrangulation of the sphere.

Sphere 10k [jpg1jpg2pdf1pdf23D pdfu3d&tex]   Sphere 30k [jpg1jpg2pdf1pdf23D pdfu3d&tex]   Sphere 50k [jpg1jpg2pdf1pdf23D pdfu3d&tex]

Uniform plane quadrangulations with 10 000 faces, 30 000 faces and 50 000 faces

Cactus embedding

Random maps are often looked at as metric spaces, where the distance between two vertices is the minimal number of edges linking them. It is not always easy to get a good grasp of this metric; a way to visualize the distances to one chosen point is to embed all the vertices at a height equal to their distance from this point. This embedding was introduced by Curien, Le Gall and Miermont in an article entitled The Brownian cactus I. Other simulations can be found in my plane quadrangulations folder but, for some mysterious reason, the results are very flat in comparison to what they should look like...

Cactus 50k [jpg1jpg2gifpdf1pdf23D pdfu3d&tex]

Cactus embedding of a uniform plane quadrangulation with 50 000 faces

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