Subdivision by bisectors is dense in the space of all triangles

TL;DR

This paper demonstrates that iteratively subdividing triangles using incenter-based bisectors produces a dense set of triangles in the space of all triangles, with specific properties about angle behavior after many subdivisions.

Contribution

It proves that repeated subdivision with the incenter can approximate any triangle arbitrarily closely and analyzes the angle behavior in typical subdivisions.

Findings

The set of subdivided triangles is dense in the space of all triangles.

Repeated subdivision can approximate any given triangle.

The smallest angle in typical subdivisions does not tend to zero.

Abstract

Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show that starting with any arbitrary triangle, the resulting set of triangles formed by this process contains triangles arbitrarily close (up to similarity) any given triangle when the point that we use to subdivide is the incenter. We also show that the smallest angle in a "typical" triangle after repeated subdivision for many generations does not have the smallest angle going to zero.