We’ve all seen images of escapetime fractals. But have you ever wondered how they form? I was playing around with sage‘s complex_plot function and got some interesting images…
For reference, here is the identity function (notice that it is also the zeroth iteration: ).
Mandelbrot Set Example
Recall that the definition of the Mandelbrot set is . (If this is notation confuses you, you may wish to refer to my previous post on anonymous functions.)
(equivalent to )
(equivalent to )
(equivalent to )
(equivalent to )
 (equivalent to )
 (equivalent to )
 (equivalent to )
 (equivalent to )

(equivalent to )
 (equivalent to )

Julia Set of 0.3 Example
 We can do this with other fractals. Recall that the Julia Set of is defined as .
 (equivalent to )
 (equivalent to )
 (equivalent to )
 (equivalent to )
 And so on…
 Update:
 Here is a far more developed and higher resolution image of the Julia set of 0.3.I don’t know what the red is.
 Here is a animation of the transition of generalised Mandelbrot set from the power of 4 to 4, inspired by this Youtube video:

Update 2:
Here are animations of the forming Mandlebrot set and Julia set respectivly:
December 20, 2009 at 03:58 
Very interesting. I made some seriously weird functions with complex exponents:
sage: complex_plot(x^(x^(x^x)),[5,5],[5,5])
sage: complex_plot(log(x^x),[5,5],[5,5])
sage: complex_plot(log(log(x^(x^x) + 0.0001)),[5,5],[5,5])
December 20, 2009 at 10:30 
Cool, recently, I played around with the same idea and made some nice interact controls for exactly the same. Maybe I should add julia, too.
About the red: red is a realpositive value. just look at your very first picture! In the julia graph at the end, white is the “escaped” infinite part, that’s usually “where the fractal is” in other plots. For me, plotting the complex values has much more fun than an escape threshold of the absolute value.
December 20, 2009 at 16:08 
For me, plotting the complex values has much more fun than an escape threshold of the absolute value.
And that’s precisely it: you can actually see what is going on.
December 23, 2009 at 22:38 
Hey chris, there are REALLY nice sets, I would absolutely love to see some more iterations with the trigonometric functions.
On a separate note, I never know you had a website! I’ll check it out when I get more proper internet/ computer conditions. I absolutely detest laptops.
Happy holidays.
December 23, 2009 at 23:04 
Sorry for replying to myself, I know it is frowned upon by the internet community, but Itai, my brother, has shown me something interesting. It is a plot of all the roots of all polynomials of 24 degrees with coefficients 1 and 1, enjoy the eye candy.
http://math.ucr.edu/home/baez/week285.html
Joyous kwanzahunnuchristmas.
February 11, 2010 at 05:34 
[...] (For more on the formation of fractals, see my previous post.) [...]
March 10, 2010 at 01:54 
[...] It’s completely different from the formation of the Mandelbrot Set! [...]