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.