Fractal curves and rugs of prescribed conformal dimension

We construct Jordan arcs of prescribed conformal dimension which are “minimal for conformal dimension,” meaning the Hausdorff and conformal dimensions are equal. 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.