Understanding Fractals, Part 4

Yesterday we put all that complex number knowledge to good use, and finally, actually constructed the Mandelbrot Set.

But so what? What’s so exciting about it? It’s just a couple weird-looking circles, right?

Well, turns out, it’s a lot more than that. Today we’ll transcend the equations and have a little fun…

Exploring the Mandelbrot Set

I’m surprised nobody yesterday asked this question: “How come we picked the number i as a ‘Tornado Number’ (i.e. inside the Mandelbrot Set) but the final graph showed it as being outside the set?”

In other words…

i plot

Mandel plot

How come the number i is green in the first picture, but blue in the second?

The answer is that i really is in the set – we were just too far away to see it. Let’s take a closer look – this time from a different rendering of the Mandelbrot. (This image and many of the following images come from right here.)

Tendril

At this higher resolution, we can see there’s actually a spidery tendril of Mandelbrottiness sneaking its way up and to the right, just barely managing to capture i in its clutches.

This tendril calls out the most fascinating feature of the Mandelbrot Set: its complexity. More precisely, its infinite complexity.

That’s right. No matter how far you zoom into the edges of this beast, you’ll always find more circles, more spirals, more microscopic worlds just as complicated as the whole big picture.

Let’s take a short journey of just eight steps. Each one merely zooms further into the one before:

zoom0

zoom1

zoom2

zoom3

zoom4

zoom5

zoom6

zoom7

There it is: buried deep, deep in the M-Set is another M-Set, almost (but not exactly) just like the big one. In fact, there are infinitely many of these little brothers, and if you were to zoom into this little one, you’d find even more along its boundaries, too.

But don’t take my word for it. Explore the M-Set for yourself.

This site is a very simple, stripped-down interface for zooming in and out wherever you want, using just the mouse wheel.

This site gives you more options. I recommend you try Manual (not Autopilot) and set the color map to Blues.

As you move in and out through the infinite complexity, try to remember that the whole thing exploded from a single, simple equation.

Well, that pretty much wraps up Fractal Week. Tell me, what did you think? Was this fun? Think you’d want to do another themed week in the future – probably on a non-math topic next time? Or just go back to random daily subjects? Let me know in the comments!

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7 responses to “Understanding Fractals, Part 4

  1. Great job Brian! I was kind of expecting to see your own code though!

  2. I am impress with how simple you made fractals Brian. Well done. Also… the pretty pictures made all the reading worth while!

  3. Pingback: Preview: Fractals You Can Draw | Ben Trube

  4. I loved the themed week! The fractals finale made it all worth it. . .Pretty pictures!
    I do have one question though: If that’s the mandelbrot set, is there any way to get a different pattern?

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