Fractal Curves and Rugs of Prescribed Conformal Dimension

TL;DR

This paper constructs specific fractal curves and rugs with a given conformal dimension, advancing understanding of minimal conformal dimension structures and potentially resolving a key conjecture in metric space theory.

Contribution

It introduces a method to construct Jordan arcs and fractal rugs with prescribed minimal conformal dimension, a novel approach in fractal geometry and conformal dimension theory.

Findings

Constructed Jordan arcs of prescribed conformal dimension

Designed fractal rugs similar to Rickman's rug with minimal conformal dimension

Potentially resolves a conjecture on metric spaces of prescribed topological conformal dimension

Abstract

We construct Jordan arcs of prescribed conformal dimension which are minimal for conformal dimension. These curves are used to design fractal rugs, similar to Rickman's rug, that are also minimal for conformal dimension. These fractal rugs could potentially settle a standing conjecture regarding the existence of metric spaces of prescribed topological conformal dimension.