Fractal geometry derived from complex bases (Q1168349)

\textit{I. Kátai} and \textit{J. Szabó} [Acta Sci. Math. 37, 255--260 (1975; Zbl 0297.12003)] proved that every Gaussian integer can be represented uniquely in base \(b\) if \(b=-n \pm i\), where \(n\) is a positive integer and the digits in the representation are \(0,1,2,\ldots,n^2\). The author investigates the subsets of the Gaussian integers and complex numbers in which the numbers can be represented in some bases \(b\) (e.g. \(b= 1 - i)\). These subsets yield some intriguing geometric patterns in the complex plane, whose boundaries are fractal curves.