Pictures / Simulations
Hyperbolic Hammersley Lines
In progress: Walking along the geodesic between the ending points in the stationary Hyperbolic Hammerley Line Model (represented in the Poincaré disk).
Simulation for symmetric lines with a Poisson Point Process of intensity 25.
Poissonian coloring of the plane
Model. On the square $[0,1]^d$, start with one blue point and one red point. Then, let points fall uniformly on the square: each time a new point arrives, it takes the color of the closest point already present. In the limit, we obtain a random coloring of the space with a fractal boundary between blue and red clusters.
Simulations. On $[0,1]^2$ with an initial red point at $(\frac{1}{3}, \frac{1}{2})$ and initial blue point at $(\frac{2}{3}, \frac{1}{2})$. Point are drawn with the associated Voronoi cells in grey and the frontier between clusters in black.
Another simulation
The same model but starting from 16 points regularly spaced on $[0,1]^2$, each one with its own color.
Tree version of the model. Now, each time a new point arrives, a whole line is drawn between this new point an the closest point of the red or blue cluster. Therefore, at all times, the red and blue cluster form trees and, in the limit, the blue and red clusters are connected.
Another simulation of the tree variant:
SIRSN
Model. A Scale-Invariant Random Spatial Network (SIRSN) is a general model of random geometry invented by Aldous (2014) and constructed in a special case by Kendall (2017) from a Poison line process on $\mathbb{R}^d$. Informally, it is the metric space is obtained by setting a speed limit on these random lines (called now roads) and defining the distance as the minimum time it takes to go between two points while following the roads of the network without going over the speed limit. This yields a random metric space with some remarkable fractal properties, some on which have been studied by Blanc (2022). In order to obtain a well-defined a scale invariant space, the speed $v$ of a road must be chosen proportional to $v^{-\gamma}$ for some exponent $\gamma > d$.
Simulation. In dimension $d=2$: roads are colored from blue to red according to their speed limit.
Confluence of geodesics. Below, the geodesics between the origin and 128 points regularly spaced on the unit disk are drawn. For small values of the exponent $\gamma$, geodesics tend to use the same very fast roads. As $\gamma$ increases to infinity, geodesics ressemble the usual lines of the euclidian metric.
Distance to the origin. The color from blue to red (heat map) represent the distance of points from the origin (located in the center of the image).
Another simulation:
Psychedelic SIRSN. Still heat maps of the distances from the origin, but with experimental color palettes...
Voronoi. The Voronoi cells for 9 points on a $3 \times 3$ grid.
