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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

arXiv:2601.13663v3cs.CGmath.CO

TL;DR

Existing results establish finitely many similarity classes and asymptotic terminal-quadruple area dominance, but not the convergence rate or precise distribution of triangle shapes. Using LEB identities, the analysis iterates left bisection until L^k(z) ∈ A, showing terminal quadruples can only arise from the corresponding LEB descendants. The area outside terminal quadruples after j bisection steps is O(ξ^j), and Theorem 1 gives the precise area distribution at every step.

  • Problem

    Existing results establish finitely many similarity classes and asymptotic terminal-quadruple area dominance, but not the convergence rate or precise distribution of triangle shapes.

    Source: §1.1 · p. 2

  • Method

    Using LEB identities, the analysis iterates left bisection until L^k(z) ∈ A, showing terminal quadruples can only arise from the corresponding LEB descendants.

    Source: Assume z ∈III · p. 10

  • Results

    The area outside terminal quadruples after j bisection steps is O(ξ^j), and Theorem 1 gives the precise area distribution at every step.

    Source: §1.2 · p. 4

  • Takeaways & Limitations

    Triangles in regions I–IV have one terminal quadruple, whereas regions V and VI contain examples with arbitrarily many.

    Source: §1.2 · p. 4

    The precise number and location of arcs and points in the proposed orbit partition remain for future research.

    Source: §4.1 · p. 19

Abstract

from arXiv · show

The Longest Edge Bisection 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 Stynes (1980) shows that for any initial triangle, the elements of this infinite family fall into finitely many similarity classes. While the set of classes is finite, it turns out that a far smaller, periodic subset of ``fat'' triangles effectively dominates the final mesh structure. This subset is comprised of periodic orbits of length four, which we refer to as {\bf terminal quadruples}. We prove the following asymptotic area distribution result: for every initial triangle, the portion of area occupied by these 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 prove 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.

1. Introduction

Prior work established that longest-edge bisection produces finitely many similarity classes and that terminal quadruples asymptotically occupy the triangle’s area. This paper seeks a quantitative account of the process, including its stepwise area distribution and the number and geometry of terminal quadruples.

1.1. Background.

  • The paper addresses the missing convergence rate and precise shape distribution left open by results on finite similarity classes and terminal-quadruple area dominance.Earlier partial results in these directions were noted in.Source: §1.1 · p. 2

1.2. Preliminaries and main results.

  • The paper models triangle similarity classes as points in a normalized domain D in the hyperbolic upper half-plane and represents bisection by maps L and R.The longest edge is scaled to length 1 and mapped to [0, 1], with the third vertex placed in the upper half-plane.Source: §1.2 · p. 2
  • A bisection graph analyzed with spectral methods supports the area-distribution result, while refined methods from are used to study orbit geometry.The paper also investigates the number of terminal quadruples and geometric properties of triangle orbits.Source: §1.2 · p. 4
  • Theorem 1 gives the precise area-distribution vector at every step and shows that the area outside terminal quadruples after j steps is O(ξ^j).This sharpens the known convergence of terminal-quadruple area to one.Source: §1.2 · p. 4
  • Theorem 2 gives q(z) = 1 for z in regions I–IV, while Theorem 3 shows that regions V and VI contain triangles with arbitrarily many terminal quadruples.Thus the number of terminal quadruples depends on the region containing the triangle.Source: §1.2 · p. 4

2. Terminal quadruples

The paper characterizes generic triangles through their orbits and defines terminal quadruples as four-element periodic configurations under the bisection maps. It also identifies the non-generic locus and its measure-theoretic status.

2.1. Generic triangles and triangles with orbits of size less than 4.

  • The paper characterizes non-generic triangles by orbit intersections with three geodesics in D.Figure 2 presents the non-generic locus.Source: §2.1 · p. 5, Figure 2 · p. 6
  • A triangle is generic exactly when its orbit avoids the three geodesics characterizing the non-generic locus.The non-generic set lies in a countable union of arcs and has measure zero, so generic triangles have full measure.Source: §2.1 · p. 5
  • A terminal quadruple is an orbit of four, not necessarily distinct, elements where L and R exchange the pairs {z1, z4} and {z2, z3}.Generic triangles have four distinct orbit elements; non-generic triangles may have repetitions.Source: §2.1 · pp. 5–6
  • For the stated point ζ, the orbit size satisfies l(z) = 1 if and only if z = ζ.The supplied corollary also specifies a condition involving a point whose displayed expression ends in 2 + 1/2i.Source: §2.1 · p. 6

2.2. The geometric structure of the terminal quadruples.

The section reports a geometric property of terminal quadruples and illustrates their placement among points of the LEB orbit. It also notes a further circular-arc pattern among selected orbit points.

2.2. The geometric structure of the terminal quadruples.

  • Every terminal quadruple lies on a Euclidean circle centered on the line Re(z) = 1/2.The four points generated from any z1 ∈ A lie on a hyperbolic circle centered at ζ = 1/2 + 1/2i.Source: §2.2 · p. 7
  • Figure 3 marks three terminal quadruples by shading the circles on which their points lie.The figure also shows points of LEB(z).Source: §2.2 · p. 7, Figure 3 · p. 7
  • The points labeled 3, 12, and 14 lie on a circle arc tangent to the real axis at 0.The text says this pattern is explained in §4.1.Source: §2.2 · p. 7

2.3. The flow towards A.

Every LEB orbit eventually reaches A, and A is exactly the locus of terminal quadruples. The proof uses a decrease in hyperbolic distance under L outside specified regions, together with the FSC theorem.

2.3. The flow towards A.

  • The argument that all orbits reach A is presented as a result not found explicitly in the cited literature.The stated approach relies on Lemma 10, which shows that repeated applications of L move points closer to A.Source: §2.3 · p. 8
  • Outside I ∪ IV, applying L strictly decreases hyperbolic distance to ζ.The proof checks the piecewise maps by region, using perpendicular bisectors to compare distances.Source: §2.3 · p. 8
  • For every z ∈ D, some iterate Ln(z) lies in A, so every LEB orbit intersects A.The proof combines the FSC theorem's finiteness of similarity classes with the distance decrease established by Lemma 10.Source: §2.3 · p. 8
  • A is exactly the locus of terminal quadruples: every point whose LEB orbit is a terminal quadruple belongs to A.The converse is stated as Corollary 12.Source: §2.3 · p. 8

2.4. Regions with q(z) = 1.

This section establishes identities for compositions of L and R in specified regions and uses region-based cases in its analysis of triangles with q(z) = 1.

2.4. Regions with q(z) = 1.

  • For z ∈ I ∪ II ∪ III, LR(z) = z, while RR(z) is a reflection across a geodesic.The reflection is given by the map z ↦ conjugate(z)/(2conjugate(z) − 1).Source: §2.4 · p. 8
  • For z ∈ I ∪ II ∪ III, R maps that region to IV, and the stated piecewise formula for L applies in IV.The proof uses these mappings in a direct computation.Source: §2.4 · p. 9
  • For z ∈ IV in the first specified case, L(z) lies in I, R(z) remains in IV, and the two displayed compositions agree.The equality is LI(LIV(z)) = LIV(RIV∪V∪VI(z)).Source: §2.4 · p. 9
  • For z ∈ IV in another specified case, the section likewise equates compositions of L and R.The stated identity is LII(LIV(z)) = LVI(RIV∪V∪VI(z)).Source: §2.4 · p. 9
  • The proof concludes q(z) = 1 for the case under discussion.The excerpt states this conclusion without including the case condition.Source: §2.4 · pp. 9–10

Assume z ∈II.

When z ∈ II, the argument uses orbit identities to conclude that q(z) = 1.

Assume z ∈II.

  • For z ∈ II, z, R(z), and RR(z) are outside terminal quadruples, while L(z) ∈ I; consequently q(z) = 1.The proof applies identities for L and R to the orbit decomposition.Source: Assume z ∈II · p. 10

Assume z ∈III.

For z in region III, terminal quadruples can arise only from descendants reached by repeated L applications. The argument ends because some iterate L^k(z) enters region A.

Assume z ∈III.

  • The triangles z, R(z), and RR(z) are not in a terminal quadruple, so any such quadruple must come from LEB(L(z)).Source: Assume z ∈III · p. 10
  • If L(z) is still in region III, the same reasoning excludes three more singletons and reduces the search to LEB(LL(z)).Source: Assume z ∈III · p. 10
  • Some iterate L^k(z) lies in region A, where the preceding cases complete the argument.Source: Assume z ∈III · p. 10

Finally, assume z ∈IV.

For z in region IV, the proof reduces the terminal-quadruple search to LEB(L(z)), which contains exactly one terminal quadruple.

Finally, assume z ∈IV.

2.5. Arbitrarily large q(z).

The section proves that the number of terminal quadruples arising from an initial triangle can be arbitrarily large. It constructs points whose iterates under h yield enough distinct triangles in region I to force that count above any prescribed bound.

2.5. Arbitrarily large q(z).

  • Theorem 3 establishes that some initial triangles produce arbitrarily many terminal quadruples.The proof constructs such triangles from points in region V.Source: §2.5 · pp. 10, 12
  • The map h is parabolic with fixed point 0, and its iterates move counterclockwise along a circle through the starting point.When Re(z) = Im(z), the circle's tangent at z is vertical.Source: §2.5 · p. 11
  • The regions B_k are shrinking copies of I, and the proof places successive h-iterates of z_m inside B_m.The induction uses the orbit relation from Lemma 17 and the geometry of the circle through z_m.Source: §2.5 · pp. 11–12
  • For each n, choosing m with 2^m ≥ n yields a point whose distinct h-iterates map into region I as distinct triangles in its LEB orbit.Each triangle in I has an orbit of length at most 4 and contributes at most two elements from I.Source: §2.5 · p. 12
  • The construction gives q(z) ≥ 2^m + 1 > n, proving that terminal-quadruple counts are unbounded.Source: §2.5 · p. 12

3. The bisection graph and its spectral properties

The section encodes longest-edge bisection as a directed graph and uses its adjacency matrix to track area across similarity classes. Spectral properties of this graph yield exponential convergence toward terminal quadruples.

3. The bisection graph and its spectral properties

  • For generic z, GLEB(z) has orbit triangles as vertices and directed L- and R-labeled edges for the two bisection maps.Source: §3 · p. 14
  • For non-generic triangles, splitting vertices resolves loops and double edges so the graph remains simple, even though this formally enlarges LEB(z).Source: §3 · pp. 14–15
  • Multiplying the adjacency matrix by a class-count vector models one bisection step, and normalized iterates encode the area distribution across similarity classes.Source: §3 · pp. 15–16
  • The eigenvalues ±2 have eigenspace dimension q(z), the number of terminal quadruples, and have no non-trivial Jordan blocks.Source: §3 · pp. 16–17
  • Terminal quadruples are precisely the strongly connected components of GLEB(z) with spectral radius 2.Source: §3 · p. 17
  • After j bisection steps, the area in triangles outside terminal quadruples is O(ξ^j), for some ξ in (0, 1).Source: §3 · p. 18

4. Concluding remarks and open problems

The concluding remarks describe geometric structure within the LEB orbit, establish continuity under small perturbations, and pose open questions about the bisection graph and terminal quadruples.

4.1. A remark on the geometric distribution of LEB(z).

  • Distinct points generated from a circle tangent to the real axis lie on another such circle and belong to distinct terminal quadruples.The passage also distinguishes the circles through points z and L(z) from circles tangent to the real axis at 0.Source: §4.1 · p. 19
  • The authors conjecture that the full orbit can be partitioned into subsets lying on arcs tangent to the real axis at 0 or 1/2, leaving their number and point locations for future analysis.Source: §4.1 · p. 19

4.2. A remark on continuity and small perturbations.

  • For every z, points in the orbit of any w within ε of z lie in the ε-neighborhood of the orbit of z.This continuity result follows from the non-increasing property of the hyperbolic metric under L and R.Source: §4.2 · p. 19
  • Nearby triangles can have sharply different complexity: the example pair has 19 versus 109 similarity classes and 1 versus 6 terminal quadruples.Source: §4.2 · pp. 19–20

4.3. Open problems.

  • Open problems ask whether the bisection graph spectrum lies in {±2, ±1, 0}, whether it is bipartite, and when its adjacency matrix is diagonalizable.Source: §4.3 · p. 20
  • The authors also ask for a formula for q(z), the number of terminal quadruples arising from an initial triangle.Source: §4.3 · p. 20
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