# The largest angle bisection procedure

## TL;DR

This paper studies an iterative triangle subdivision method based on bisecting the largest angles, proving convergence properties and the diversity of resulting small triangles.

## Contribution

It introduces and analyzes the largest angle bisection procedure, establishing convergence and diversity results for the generated triangle partitions.

## Key findings

- Diameters of subdivided triangles tend to zero as iterations increase.
- Smallest angles in the partitions are bounded away from zero.
- Number of non-congruent small triangles becomes unbounded, except for a specific initial triangle.

## Abstract

The {\it largest angle bisection} procedure is the operation which partitions a given triangle, $T$, into two smaller triangles by constructing the angle bisector of the largest angle of $T$. Applying the procedure to each of these two triangles produces a partition of $T$ into four smaller triangles. Continuing in this manner, after $n$ iterations, the initial triangle is divided into $2^n$ small triangles. We prove that as $n$ approaches infinity, the diameters of all these $2^n$ triangles tend to $0$, the smallest angle of all these triangles is bounded away from $0$, and that, with the exception of $T$ being an isosceles right triangle, the number of dissimilar triangles is unbounded.