Larry Crone

Associate Professor Emeritus
Department of Mathematics and Statistics, American University
Alum Bank, PA

For the past thirty years I have been using color to display complex valued functions. Such plots clearly show features such as zeros, poles, and branch cuts. I have also long been interested in the inverse Schroeder function, which is related to functional commutivity and the iteration of functions. I soon put the two together and discovered that it is particularly straight forward to plot the inverse Schroeder function corresponding to rational functions with a repelling fixed point at zero. Although these are all meromorphic, they are extremely complicated and their plots are often mistaken for fractals. I have investigated the plots of many inverse Schroeder functions, and have been overwhelmed by the beauty and variety of the images.

19 x 31
19 x 31
13 in x 16 in

In addition to its beauty, this image is of interest because the underlying quadratic rational function has an attracting fixed point cycle of order 19, and another of order 31. Just as a mountain presents many targets to a photographer, this function can be viewed from different perspectives, and it was hard to decide which one to use. The windows program Gplot, the camera which took this picture, is available for free download at