  In past decades, the application of fractals to electrode design for enhanced
signaling and electrochemical performance was a popular subject and enabled the
growth of consumer micro-electronics. Supercapacitors, which are energy storage
devices with many promising characteristics, have largely grown alongside of
such developments in electronics, but little work has been done to use fractal
electrodes in supercapacitors. In this work, plane-filling and fractal patterns
were used in designing laser scribed graphene supercapacitor electrodes,
allowing the scaling laws of capacitance with respect to fractal order and
complexity to be examined for the first time. An interesting exponential
relationship between capacitance and fractal order for the more open structured
fractals was observed, the exponent of which was proportional to the Hausdorff
dimension. Additional non-linear relationships between capacitance and order
were observed for other structures which was correlated with inter-plate
repulsion and differences in path length. These findings provide the first step
in maximizing the efficiency of fractal-based electrolytic devices by exploring
the non-intuitive trends in capacitance with respect to fractal order and
complexity.
