## Formation of Escape-Time Fractals

We’ve all seen images of escape-time 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: $f^0(x)=x$).

Identity Function

### Mandelbrot Set Example

Recall that the definition of the Mandelbrot set is $\left\{x|x\in \mathbb{C}; \lim_{n\to\infty}(z\to z^2+x)^n(x)\neq\pm\infty\right\}$. (If this is notation confuses you, you may wish to refer to my previous post on anonymous functions.)

Iteration 1

$(z\to z^2+x)^1(x)$ (equivalent to $x^2+x$)

Iteration 2

$(z\to z^2+x)^2(x)$ (equivalent to $(x^2+x)^2+x$)

Iteration 3

$(z\to z^2+x)^3(x)$ (equivalent to $((x^2+x)^2+x)^2+x$)

Iteration 4

$(z\to z^2+x)^4(x)$ (equivalent to $(((x^2+x)^2+x)^2+x)^2+x$)

Iteration 5

$(z\to z^2+x)^5(x)$ (equivalent to $((((x^2+x)^2+x)^2+x)^2+x)^2+x$)

Iteration 6

$(z\to z^2+x)^6(x)$ (equivalent to $(((((x^2+x)^2+x)^2+x)^2+x)^2+x)^2+x$)

Iteration 7

$(z\to z^2+x)^7(x)$ (equivalent to $((((((x^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)^2+x$)

Iteration 8

$(z\to z^2+x)^8(x)$ (equivalent to $(((((((x^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)^2+x$)

Iteration 9

$(z\to z^2+x)^9(x)$ (equivalent to $((((((((x^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)^2+x$)

Iteration 10

$(z\to z^2+x)^10(x)$ (equivalent to $(((((((((x^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)^2+x)+x$)

### Julia Set of 0.3 Example

We can do this with other fractals. Recall that the Julia Set of $0.3$ is defined as $\left\{x|x\in \mathbb{C}; \lim_{n\to\infty}(z\to z^2+0.3)^n(x)\neq\pm\infty\right\}$.

Iteration 1 of the Julias Set of 0.3

$(z\to z^2+0.3)^1(x)$  (equivalent to $x^2+0.3$)

Iteration 2 of the Julia Set of 0.3

$(z\to z^2+0.3)^2(x)$  (equivalent to $(x^2+0.3)^2+0.3$)

Iteration 3 of the Julia Set of 0.3

$(z\to z^2+0.3)^3(x)$  (equivalent to $((x^2+0.3)^2+0.3)^2+0.3$)

Iteration 4 of the Julia Set of 0.3

$(z\to z^2+0.3)^4(x)$  (equivalent to $(((x^2+0.3)^2+0.3)^2+0.3)^2+0.3$)
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.

Complex plot of the Julia Set of 0.3

Here is a animation of the transition of generalised Mandelbrot set from the power of -4 to 4, inspired by this Youtube video:

Generalized Mandlebrot from the power of -4 to 4

Update 2:

Here are animations of the forming Mandlebrot set and Julia set respectivly:

Formation of the Mandlebrot Set

The Formation of the Julia Set

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### 7 Responses to “Formation of Escape-Time Fractals”

1. Vitalik Buterin Says:

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])

2. schilly Says:

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 real-positive 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.

• christopherolah Says:

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.

3. Assaf Says:

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.

• Assaf Says:

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.

4. Fractals with non-Integer powers « Christopher Olah’s Blog Says:

[...] (For more on the formation of fractals, see my previous post.) [...]

5. Why are there mini sets in the Mandelbrot Set? « Christopher Olah's Blog Says:

[...] It’s completely different from the formation of the Mandelbrot Set! [...]