Here’s a bit more on the beauty of roots—some things that may have escaped those of you who weren’t following this blog carefully!
Greg Egan has a great new applet for exploring the roots of Littlewood polynomials of a given degree—meaning polynomials whose coefficients are all ±1:
• Greg Egan, Littlewood applet.
Move the mouse around to create a little rectangle, and the applet will zoom in to show the roots in that region. For example, the above region is close to the number -0.0572 + 0.72229i.
Then, by holding the shift key and clicking the mouse, compare the corresponding ‘dragon’. We get the dragon for some complex number by evaluating all power series whose coefficients are all ±1 at this number.
You’ll see that often the dragon for some number resembles the set of roots of Littlewood polynomials near that number! To get a sense of why, read Greg’s explanation. However, he uses a different, though equivalent, definition of the dragon (which he calls the ‘Julia set’).
He also made a great video showing how the dragons change shape as you move around the complex plane:
The dragon is well-defined for any number inside the unit circle, since all power series with coefficients ±1 converge inside this circle. If you watch the video carefully—it helps to make it big—you’ll see a little white cross moving around inside the unit circle, indicating which dragon you’re seeing.
I’m writing a paper about this stuff with Dan Christensen and Sam Derbyshire… that’s why I’m not giving a very careful explanation now. We invited Greg Egan to join us, but he’s too busy writing the third volume of his trilogy Orthogonal.
Azimuth 
I’m not really sure how these images from A New Kind of Science were generated, but there’s a definite resemblance.
It looks like the same general idea of parameterising an iterated function system.
I was unable to see that page on A New Kind of Science—I got a message saying cookies must be enabled, even after I enabled cookies. I checked that I don’t have http://www.wolframscience.com blocked… and at that point I ran out of energy for this particular task. Unless Greg tells me it’s something I should study (and I think he’s not telling me that), I’ll let it go at that.
If it’s from “A New Kind of Science” it’s probably about cellular automata.
Ah, I got the same message about cookies, but I have a paper copy of the book so I went and looked at that, instead of battling the web site’s quirks!
The figure in question is from a chapter about substitution systems (L-systems) and plant growth, with no explicit discussion of functions of a complex variable. I’m sure it could all be translated into the language of iterated function systems, but the particular functions would be different from the ones we’re studying with the Littlewood polynomials.
“We invited Greg Egan to join us, but he’s too busy writing the third volume of his trilogy Orthogonal.”
What happened to the 2nd book?
It must be written but not printed yet. The third volume is due in December, he said. I’d be terrified if I were writing novels with a deadline!
[…] Sam Derbyshire and I are writing a paper on a fun topic I’ve discussed here before: the beauty of roots. We’re getting help from Greg Egan, but he’s too busy writing his next novel to commit […]