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Towards Constructing Geodesic Nets with Four Boundary Vertices and an Increasing Number of Balanced Vertices

Fabian Parsch, Hanrui Zhang

TL;DR

The paper tackles whether geodesic nets in the Euclidean plane with four unbalanced (boundary) vertices can host an unbounded number of balanced vertices. It provides a constructive answer by presenting an irreducible net with four boundary vertices and $25$ balanced vertices, exceeding the prior known maximum of $16$, and introduces irregular, non-symmetric balance features along with center-imbalance cancellation. The construction rests on a dodecagon-centered topology augmented by Fermat-point based interior and boundary vertices, governed by carefully chosen angle parameters $(oldsymbol{eta},oldsymbol{eta})$ that satisfy balancing equations. Beyond the explicit example, the work outlines methods—irregular vertices, center balance, gradient-descent search—that may generalize to a series of nets, potentially supporting the conjecture that the number of balanced vertices is unbounded for nets with four boundary vertices. This advances understanding of Gromov-type questions for four unbalanced vertices and offers a concrete path toward proving nonexistence of universal upper bounds in this regime.

Abstract

We construct a geodesic net in the plane with four boundary (unbalanced) vertices that has 25 balanced vertices and that is irreducible, i.e. it does not contain nontrivial subnets. This net is novel and remarkable for several reasons: (1) It increases the previously known maximum for balanced vertices of nets of this kind from 16 to 25. (2) It is, to our knowledge, the first such net that includes balanced vertices whose incident edges are not exhibiting symmetries of any kind. (3) The approach taken in the construction is quite promising as it might have the potential for generalization. This would allow to construct a series of irreducible geodesic nets with four boundary vertices and an arbitrary number of balanced vertices, answering a conjecture that the number of balanced vertices is in fact unbounded for nets with four boundary vertices. This would stand in stark contrast to the previously proven theorem that for three boundary vertices, there can be at most one single balanced vertex.

Towards Constructing Geodesic Nets with Four Boundary Vertices and an Increasing Number of Balanced Vertices

TL;DR

The paper tackles whether geodesic nets in the Euclidean plane with four unbalanced (boundary) vertices can host an unbounded number of balanced vertices. It provides a constructive answer by presenting an irreducible net with four boundary vertices and balanced vertices, exceeding the prior known maximum of , and introduces irregular, non-symmetric balance features along with center-imbalance cancellation. The construction rests on a dodecagon-centered topology augmented by Fermat-point based interior and boundary vertices, governed by carefully chosen angle parameters that satisfy balancing equations. Beyond the explicit example, the work outlines methods—irregular vertices, center balance, gradient-descent search—that may generalize to a series of nets, potentially supporting the conjecture that the number of balanced vertices is unbounded for nets with four boundary vertices. This advances understanding of Gromov-type questions for four unbalanced vertices and offers a concrete path toward proving nonexistence of universal upper bounds in this regime.

Abstract

We construct a geodesic net in the plane with four boundary (unbalanced) vertices that has 25 balanced vertices and that is irreducible, i.e. it does not contain nontrivial subnets. This net is novel and remarkable for several reasons: (1) It increases the previously known maximum for balanced vertices of nets of this kind from 16 to 25. (2) It is, to our knowledge, the first such net that includes balanced vertices whose incident edges are not exhibiting symmetries of any kind. (3) The approach taken in the construction is quite promising as it might have the potential for generalization. This would allow to construct a series of irreducible geodesic nets with four boundary vertices and an arbitrary number of balanced vertices, answering a conjecture that the number of balanced vertices is in fact unbounded for nets with four boundary vertices. This would stand in stark contrast to the previously proven theorem that for three boundary vertices, there can be at most one single balanced vertex.

Paper Structure

This paper contains 16 sections, 6 theorems, 13 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Construction of the Net
  3. Conventions
  4. Overview
  5. The Two Angles alpha and beta
  6. Construction of Inner Dodecagon:
  7. Construction of the Balanced Vertices in the Interior of the Dodecagon
  8. Construction of the Four Boundary Vertices
  9. Construction of Additional Balanced Vertices
  10. Proof of irreducible geodesic net properties
  11. Towards a series of nets
  12. Using "irregular" vertices
  13. "Canceling" imbalance at the centre
  14. Using gradient descent to find candidate nets
  15. Attempts at generalizing the construction
  16. ...and 1 more sections

Key Result

Theorem 1.4

Each geodesic net with three unbalanced vertices (of arbitrary degree) on the plane endowed with a Riemannian metric of non-positive curvature has at most one balanced vertex. (Furthermore, this statement is not true if we allow for positive curvature.)

Figures (7)

  • Figure 1.1: Example of a geodesic net with 4 boundary vertices and 27 balanced vertices on the Euclidean plane with the boundary vertices highlighted, as constructed in Section 7 of Parsch2018-nu. Despite its seemingly complex structure, it is just an overlay of 7 simpler nets as seen on the right.
  • Figure 1.2: Two irreducible geodesic nets with 4 boundary vertices. At the top, the example with 16 balanced vertices, constructed in Parsch2019-fz. At the bottom, the example with 25 balanced vertices that we construct in the present paper. Note that in both cases, some edges may seem to coincide due to angles being extremely small. This is why some angles are slightly exaggerated in the zoom-ins.
  • Figure 2.1: The dodecagon defined at the beginning of the construction
  • Figure 2.2: Construction of interior vertices $p$ and $f_{2}$ as well as the boundary vertex $d_2$ with relevant connections to scale. The angle $\beta$ is labeled.
  • Figure 2.3: Construction of vertices $c_{2}$ and $e_{2}$ with relevant connections to scale.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.4
  • Definition 1.5
  • Conjecture 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 5 more