Fractional quantum Hall effects as an example of fractal geometry in nature

A prescription is provided for constructing the Hall curve including both integral (I)- and fractional (F)-quantum Hall effects (QHE) that is based upon the iterative application of particular transformations simultaneously to the Hall resistance (R(xy)) and magnetic field (B) axes of a template constructed from the elementary (integral quantum) Hall curve to filling factor nu = 1. The construction shows that scaled copies of the elementary Hall curve reappear in various parts of the constructed curve upon increasing the magnification, resulting in FQHE sequences in higher Landau bands, and novel FQHE sequences between main sequence FQHE's in the lowest Landau band. The self similarity observed in the constructed Hall curve helps to draw a connection between FQHE's and the classical problem of an electron-in-a-periodic-potential-subjected-to-a-magnetic-field ('Hofstadter's butterfly'), and suggests that fractional quantum Hall effects constitute another manifestation of fractal geometry in nature-one that might also be viewed as a signature of transport in a Wigner crystal.