Renormalisation of curlicues

. . . is the title of a paper by Berry and Goldberg from 1988. Berry and Goldberg's curlicues are a family of curves containing near-copies of themselves. You can see them by bringing up the curlicues applet window from this page.

There used to be a Java applet here, but browsers don't allow those any more. However, you can download the Curli.jar file and run it as a standalone program. The Curli program is written in literate Java and is free software.

Curlicues are defined by

Sum [n from 1 to L]   exp(i pi tau n^2)

where tau is a parameter between 0 and 1. As you can see, each term in the sum is a unit vector in the complex plane. Join up all these unit vectors nose-to-tail and look at the pattern from a large distance, and you have a curlicue. Physicists will recognise a curlicue as a discrete version of Cornu's spiral in optics, and indeed curlicues can be interpreted as diffraction experiments too.

Remarkably, a curlicue of length L always starts off with a small near-copy of itself. The renormalise operation takes this near copy and scales, rotates, and reflects it to show the resemblance. (To overlay, change colour before pressing the renormalize button. You can also zoom in and out, and re-centre by clicking within the plot.) Notice how renormalising keeps removing the smallest level of structure.

The numbers below the buttons are tau, its continued fraction, and L. You can specify any of these numbers by typing and then pressing return. The decimal number and the continued fraction are slightly inconsistent, because of roundoff errors, but that won't make any visible difference to the graphics. Try playing with different continued fractions and see which ones produce the most beautiful curlicues.

The details of the renormalisation operation are given in the Berry-Goldberg paper, but basically it deletes the first term in the continued fraction from tau. Which tells us that the higher terms in the continued fraction specify the small scales of the curlicue, while the deeper terms in the continued fraction specify the large scales of the curlicue.

I like to think of curlicues as turning a number inside out.