Details

A curlicue with parameters and is the path in the complex plane obtained by connecting the partial sums of

,

or, equivalently, the polygonal path in the plane obtained by starting at the origin, followed by unit steps in directions .

Assuming (without loss of generality) that , Berry and Goldberg [1] showed that if and, if is even, , otherwise , then

(1) if is odd

and

(2) if is even,

where denotes the complex conjugate of . The transformation given by (1) and (2) relates a curlicue with parameters and to a curlicue with parameters and scaled by a factor . This transformation is called the renormalization of the curlicue . This Demonstration illustrates the renormalization process by showing the curlicues on the left and right sides of (1) and (2) for all parameter pairs with and , as well as for a set of interesting predefined values of and .

Reference

[1] M. V. Berry and J. Goldberg, "Renormalization of Curlicues," Nonlinearity, 1(1), 1988 pp. 1–26. doi:10.1088/0951-7715/1/1/001.