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On the stability, complexity, and distribution of similarity classes of the longest edge bisection process for triangles

Daniel Kalmanovich, Yaar Solomon

TL;DR

The paper analyzes the Longest Edge Bisection (LEB) process for triangles through a hyperbolic moduli-space model, revealing that the infinite orbit of any initial triangle splits into a finite set of similarity classes and, more dominantly, into a small, stable collection called terminal quadruples. It proves an exponential convergence of area to terminal quadruples and provides exact distributions for how area is allocated across these classes via even/odd limits, $w_{even}$ and $w_{odd}$. A bisection graph $\mathrm{G_{LEB}}(z)$ is introduced and studied spectrally, showing that the spectral radius $2$ corresponds to terminal quadruples and that their count $q(z)$ governs the dimensionality of the corresponding eigenspaces. The work also establishes that regions $\text{I}$–$\text{IV}$ yield $q(z)=1$, while $q(z)$ can be arbitrarily large in regions $\text{V}$–$\text{VI}$, and proves several geometric properties of terminal quadruples, laying groundwork for a full geometric distribution of the LEB orbit. Overall, the paper advances both the stability and complexity understanding of LEB meshes, with implications for mesh quality and adaptive refinement strategies in computational geometry and numerical analysis.

Abstract

The Longest Edge Bisection (LEB) of a triangle is performed by joining the midpoint of its longest edge to the opposite vertex. Applying this procedure iteratively produces an infinite family of triangles. Surprisingly, a classical result of Adler (1983) shows that for any initial triangle, this infinite family falls into finitely many similarity classes. While the set of classes is finite, we show that a far smaller, stable subset of ``fat'' triangles, called {\bf terminal quadruples}, effectively dominates the final mesh structure. We prove the following asymptotic area distribution result: for every initial triangle, the portion of area occupied by terminal quadruples tends to one, with the convergence occurring at an exponential rate. In fact, we provide the precise distribution of triangles in every step. We introduce the {\bf bisection graph} and use spectral methods to establish this result. Given this dominance, we provide a complete characterization of triangles possessing a single terminal quadruple, while conversely exhibiting a sequence of triangles with an unbounded number of terminal quadruples. Furthermore, we reveal several fundamental geometric properties of the points of a terminal quadruple, laying the groundwork for studying the geometric distribution of the entire orbit. Our analysis leverages the hyperbolic geometry framework of Perdomo and Plaza (2014) and refines their techniques.

On the stability, complexity, and distribution of similarity classes of the longest edge bisection process for triangles

TL;DR

The paper analyzes the Longest Edge Bisection (LEB) process for triangles through a hyperbolic moduli-space model, revealing that the infinite orbit of any initial triangle splits into a finite set of similarity classes and, more dominantly, into a small, stable collection called terminal quadruples. It proves an exponential convergence of area to terminal quadruples and provides exact distributions for how area is allocated across these classes via even/odd limits, and . A bisection graph is introduced and studied spectrally, showing that the spectral radius corresponds to terminal quadruples and that their count governs the dimensionality of the corresponding eigenspaces. The work also establishes that regions – yield , while can be arbitrarily large in regions –, and proves several geometric properties of terminal quadruples, laying groundwork for a full geometric distribution of the LEB orbit. Overall, the paper advances both the stability and complexity understanding of LEB meshes, with implications for mesh quality and adaptive refinement strategies in computational geometry and numerical analysis.

Abstract

The Longest Edge Bisection (LEB) of a triangle is performed by joining the midpoint of its longest edge to the opposite vertex. Applying this procedure iteratively produces an infinite family of triangles. Surprisingly, a classical result of Adler (1983) shows that for any initial triangle, this infinite family falls into finitely many similarity classes. While the set of classes is finite, we show that a far smaller, stable subset of ``fat'' triangles, called {\bf terminal quadruples}, effectively dominates the final mesh structure. We prove the following asymptotic area distribution result: for every initial triangle, the portion of area occupied by terminal quadruples tends to one, with the convergence occurring at an exponential rate. In fact, we provide the precise distribution of triangles in every step. We introduce the {\bf bisection graph} and use spectral methods to establish this result. Given this dominance, we provide a complete characterization of triangles possessing a single terminal quadruple, while conversely exhibiting a sequence of triangles with an unbounded number of terminal quadruples. Furthermore, we reveal several fundamental geometric properties of the points of a terminal quadruple, laying the groundwork for studying the geometric distribution of the entire orbit. Our analysis leverages the hyperbolic geometry framework of Perdomo and Plaza (2014) and refines their techniques.
Paper Structure (13 sections, 19 theorems, 29 equations, 11 figures, 1 table)

This paper contains 13 sections, 19 theorems, 29 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries and main results
  4. Terminal quadruples
  5. Generic triangles and triangles with orbits of size less than $4$
  6. The geometric structure of the terminal quadruples
  7. Regions with $q(z)=1$
  8. Arbitrarily large $q(z)$
  9. The bisection graph and its spectral properties
  10. Concluding remarks and open problems
  11. A remark on the geometric distribution of $\mathrm{LEB}(z)$
  12. A remark on continuity and small perturbations
  13. Open problems

Key Result

Theorem 1

For a triangle $z$ and $j\in\mathbb{N}$ let $w_j\in \mathbb{R}^{l(z)}$ be the probability vector that describes the partition of the area of $z$ into triangles of different similarity classes, after $j$ steps. Then, for every triangle $z$ we have the following:

Figures (11)

  • Figure 1: The normalized region $D$
  • Figure 2: The subdivision of $D$ into $6$ subregions
  • Figure 4: The non-generic triangles in $D$: The triangles on the upper red curve satisfy $z=L(z)$, the triangles on the lower red curve satisfy $z=R(z)$. The triangles on the green curve satisfy $L(z)=R(z)$.
  • Figure 5: The non-generic triangles in the shaded terminal region $A$: the blue$\zeta$ generates $1$ triangle. The triangles on the red curves generate $2$ triangles. The triangles on the green curves generate $3$ triangles.
  • Figure 7: The points of $\mathrm{LEB}(z)$ for $z=\frac{1}{9}+\frac{1}{7}i$. There are $3$ terminal quadruples, and the circles that they lie on are shaded.
  • ...and 6 more figures

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Proposition 5
  • proof
  • Remark 6
  • Corollary 7
  • proof
  • Proposition 8
  • ...and 28 more