On the Steiner tree connecting a fractal set

TL;DR

This paper constructs a unique self-similar binary tree that solves the Steiner problem for a specific fractal set, connecting the set of leaves with minimal total length.

Contribution

It provides the first explicit example of a self-similar Steiner tree connecting a fractal boundary, demonstrating existence and uniqueness.

Findings

Constructed an explicit self-similar Steiner tree for a fractal set

Proved the tree is essentially unique as a minimal connection

Showed the tree's boundary coincides with the fractal set

Abstract

We construct an example of an infinite planar embedded self-similar binary tree $\Sigma$ which is the essentially unique solution to the Steiner problem of finding the shortest connection of a given planar self-similar fractal set $C$ of positive Hausdorff dimension. The set $C$ can be considered the set of leaves, or the ``boundary``, of the tree $\Sigma$, so that $\Sigma$ is an irreducible solution to the Steiner problem with datum $C$ (i.e. $\Sigma\setminus C$ is connected).