# An example for a nontrivial irreducible geodesic net in the plane

## TL;DR

This paper constructs a complex geodesic net in the Euclidean plane with four boundary vertices, demonstrating a nontrivial, irreducible structure that is not a tree, advancing understanding of geodesic nets.

## Contribution

It provides the first example of a nontrivial, irreducible geodesic net with four boundary vertices in the plane that is not a tree.

## Key findings

- Constructed a geodesic net with 4 boundary vertices and 16 balanced vertices.
- Proved the net is irreducible and contains no proper geodesic subnets.
- First example of such a nontrivial geodesic net in the Euclidean plane.

## Abstract

We construct a geodesic net in the plane with four unbalanced (boundary) vertices that has 16 balanced vertices and does not contain proper geodesic subnets. This is the first example of an irreducible geodesic net in the Euclidean plane with 4 boundary vertices that is not a tree.