This Demonstration shows pseudorandom walks constructed from generalized Gauss sums, defined by , where the modulus and the exponent are integers of at least 2. (The case reduces to the classical quadratic Gauss sum.)

The random walks start at the origin, then the end point, as indicated by the yellow dot, takes steps given by the terms of the sum. The walks exhibit complicated behavior with curlicue patterns and sometimes unexpected symmetries.

(Based on an undergraduate research project at the Illinois Geometry Lab by Ka Fung Tjin, Erqian Wang and Yifan Zhang and directed by A. J. Hildebrand.)

This Demonstration is inspired by visualizations from [1].

[1] R. R. Moore and A. J. van der Poorten, "On the Thermodynamics of Curves and Other Curlicues," Miniconference on Geometry and Physics (M. N. Barber and M. K. Murray, eds.), Canberra, Australia: The Australian National University, 1989 pp. 82–109. maths.anu.edu.au/files/CMAProcVol22-MoorePoorten.pdf.