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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