Fractal Curves Generated by the Sum-of-Digits Function
Fractal-like curves are generated by the sum-of-digit function for a given base with two additional parameters and . In some special cases, these curves reduce to known fractals; for example, when and , the curve is a scaled version of the famous Koch curve. For general parameters and , the curves can take on a remarkable and largely unpredictable variety of shapes, from distinctive geometric shapes to fractal-like patterns to random clouds exhibiting no apparent pattern.
Contributed by: Adithya Swaminathan, Dimitrios Tambakos and A. J. Hildebrand (June 2021)
(Based on an undergraduate research project at the Illinois Geometry Lab in Spring 2021)
Open content licensed under CC BY-NC-SA
Let be the base- representation of . The the sum-of-digits function of in base is defined to be .
The curves shown in this Demonstration are constructed as follows. Fix an integer and two parameters and , and let
Let be the polygonal curve that starts at the origin and whose steps are unit vectors in the direction of the angles , . Thus, is the infinite curve obtained by connecting the partial sums
interpreted as points in the complex plane (with the empty sum interpreted as ).
There are several special cases that are of particular interest:
• When and , the curve is a scaled version of the Koch curve . The point at step is , where is the Thue–Morse sequence, defined by if the sum of binary digits of is even, and otherwise.
• When , is an integer and is congruent to or modulo , the curve is periodic and represents a finite polygonal curve that can be explicitly described .
• When , the angles reduce to and so do not depend on the sum-of-digit function. The resulting curves are known as curlicue fractals .
• When and are integers, the curve encodes the behavior of the sum-of-digit function , with restricted to an arithmetic progression modulo , in residue classes modulo [1, 4].
• For general parameters and , the behavior of is more mysterious, and very little is known of such cases.
 J.-P. Allouche and G. Skordev, "Von Koch and Thue–Morse Revisited," Fractals, 15(4), 2007 pp. 405–409. doi:10.1142/S0218348X07003630.
 D. H. Lehmer and E. Lehmer, "Picturesque Exponential Sums, I," The American Mathematical Monthly, 86(9), 1979 pp. 725–733. doi:10.1080/00029890.1979.11994899.
 F. M. Dekking and M. Mendès France, "Uniform Distribution Modulo One: A Geometrical Viewpoint," Journal für die reine und angewandte Mathematik, 1981(329), 1981 pp. 143–153. doi:10.1515/crll.1981.329.143.
 F. M. Dekking, "On the Distribution of Digits in Arithmetic Sequences," Séminaire de théorie des nombres de Bordeaux (1982–1983), Bordeaux: Société Arithmétique de Bordeaux, 1982 pp. 1–12. www.jstor.org/stable/44166422.