The regular pentagon cannot tile the plane, although many polygons related to the pentagon can tile the plane, including those that admit Penrose tilings. Plane-filling fractal curves are closely-related to tilings, and so the question is asked: are there pentagon-based fractal curves that can fill the plane? This paper introduces a method for designing fractal curves with local five-fold rotational symmetry, using a special mathematical tool: the first 10th root of unity—a complex number that easily generates the golden ratio. It can be raised to integer powers for rotation, and can be scaled by its own derived golden ratio, which itself can be raised to integer powers to determine the fractal scaling factor. Building upon previous work in categorizing fractal curves with complex integers, this paper incorporates inspiration from some new developments in tiling and fractals.