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

TL;DR

This paper constructs a novel geodesic net with four boundary vertices and 25 balanced vertices, surpassing previous limits, and introduces a promising approach for creating nets with arbitrarily many balanced vertices, addressing a longstanding conjecture.

Contribution

It presents the first geodesic net with four boundary vertices having more than 16 balanced vertices and introduces a method that could generate infinitely many such nets.

Findings

Maximum balanced vertices increased from 16 to 25

First net with non-symmetric incident edges at balanced vertices

Potential for generalization to unbounded balanced vertices

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.