Pickover biomorphs and non-standard complex numbers. (August 2018) - Jakubska-Busse, A.; Janowicz, M.W.; Ochnio, L.; Ashbourn, J.M.A.
Highlights: The concept of Pickover biomorphs extended to the set of complex numbers is proposed. The graphical structures which emerge from iterations of complex functions are simpler. Moran's correlation function, sample entropy and the Hurst exponent are useful in the analysis of biomorphs. The dependence of sample entropy appears to be complementary. Abstract: In this study Pickover biomorphs are analysed as being dependent on the chosen complex number system in which iterations of analytic functions are performed. Moran's spatial autocorrelation function and two forms of entropy, the Shannon entropy and the sample entropy, are chosen in order to find correlations and measure complexity in Pickover biomorphs. These turn out to be strongly correlated and low-entropy objects with a fractal dimension between 1.4 and 2. It is shown that there is a strong maximum in correlation and a strong minimum in entropy for the case of Galilean complex numbers corresponding to the square of the generalised imaginary unit being equal to zero.