Gluing circles
What is this?
This image is the result of a random iterative gluing procedure, performed
with a sequence of circles whose circumferences are given by the sequence
n-3/5. Starting with one circle, we add a new circle to the
construction at each step, by identifying one of its point to a point
randomly chosen on the circles that are already present. The way of
choosing this random point is the following: first, we choose one of the
circles, where for any k, the kth one in order of creation is chosen with
probability proportional to k -3/2, and then we choose a uniform random
point on that circle.
The almost sure Hausdorff dimension of the structure that we obtain after an
infinite number of steps is given by the surprising formula
10⁄3 - sqrt(5).
Reference
D. Sénizergues.
Random gluings of metric spaces. Ann. Probab. 47, 3812-3865, (2019).