On barycentric subdivision, with simulations

TL;DR

This paper studies a Markov chain generated by barycentric subdivision of triangles, showing that triangles become flatter over time, with angles approaching pi, and analyzes the limit behavior through probabilistic methods and simulations.

Contribution

It introduces a probabilistic analysis of barycentric subdivision, characterizing the limit set and angle convergence, and identifies the stationary distribution of an associated Markov chain.

Findings

Triangles become flatter with isoperimetric values diverging

Largest angle converges to pi in probability

Limit set of the opposite vertex is segment [0,1/2]

Abstract

Consider the barycentric subdivision which cuts a given triangle along its medians to produce six new triangles. Uniformly choosing one of them and iterating this procedure gives rise to a Markov chain. We show that almost surely, the triangles forming this chain become flatter and flatter in the sense that their isoperimetric values goes to infinity with time. Nevertheless, if the triangles are renormalized through a similitude to have their longest edge equal to $[0,1]\subset\CC$ (with 0 also adjacent to the shortest edge), their aspect does not converge and we identify the limit set of the opposite vertex with the segment [0,1/2]. In addition we prove that the largest angle converges to $\pi$ in probability. Our approach is probabilistic and these results are deduced from the investigation of a limit iterated random function Markov chain living on the segment [0,1/2]. The stationary distribution of this limit chain is particularly important in our study. In an appendix we present related numerical simulations (not included in the version submitted for publication).