# Renormalization of Curlicue Fractals

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Given a positive integer and a real number in the interval , consider the polygonal path in the plane starting at the origin, followed by steps of length 1 in directions , respectively. Such a path is called a curlicue or curlicue fractal, since it exhibits a fractal-like structure. Berry and Goldberg showed that a curlicue with parameter pair is (up to scaling and rotation/reflection) similar to a curlicue with parameter pair , where and are given by and if is even, if is odd. Here denotes the fractional part of . Transformation of the former curlicue into the latter is called renormalization. It is of considerable interest, since it can rigorously prove some geometric features of curlicues. This Demonstration illustrates the renormalization process by showing the original and renormalized curlicues for a wide range of parameters and .

Contributed by: Tri Do, Kevin Liu and A. J. Hildebrand (August 27)

(Based on an undergraduate research project at the Illinois Geometry Lab in Fall 2022)

Open content licensed under CC BY-NC-SA

## 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.

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