The Nonlinear Library LW - An Introduction To The Mandelbrot Set That Doesn't Mention Complex Numbers by Yitz
Jan 17, 2024
11:04
Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: An Introduction To The Mandelbrot Set That Doesn't Mention Complex Numbers, published by Yitz on January 17, 2024 on LessWrong.
Note: This post assumes you've heard of the Mandelbrot set before, and you want to know more about it, but that you find imaginary and complex numbers (e.g. the square root of negative one) a bit mystifying and counterintuitive. Instead of helping you understand the relevant math like a reasonable person would, I'm just going to pretend the concept doesn't exist, and try to explain how to generate the Mandelbrot set anyway.
My goal is for this post to (theoretically) be acceptable to the historical René Descartes, who coined the term "Imaginary number" because he did not believe such things could possibly exist.
I hereby formally invite you to a dance.
Since we're (presumably) both cool, hip people, let's go with a somewhat avant-garde dance that's popular with the kids these days.
I call this dance the Mandelbrot Waltz, but you can call it whatever you'd like. This dance follows very simple rules, with the quirk that your starting location will influence your part in the dance. You will unfortunately be cursed to dance forever (there's always a catch to these dance invitations!), but if you ever touch the edges of the dance floor, the curse will be lifted and your part in the dance ends, so it's really not all that bad...
In case you don't already know the moves, I'll describe how to do the dance yourself (if given an arbitrary starting point on the dance floor) step-by-step.
How To Perform The Mandelbrot Waltz: A Step-By-Step Guide
Preparation:
You will need: Yourself, an empty room, and a drawing tool (like chalk or tape).
Setup: Draw a line from the center of the room to the nearest part of the wall, like so:
Now, draw a circle around the room's center, such that it intersects the "orienting line" halfway through. It should look something like this:
Starting Position:
Choose a starting point anywhere you want in the room. Remember this position - or jot it down on a notepad if your memory is bad - for later.
Step 1 - Rotation Doubling:
Imagine a line connecting your current position to the center of the circle:
Find the orienting line we drew on the floor earlier, and measure, counterclockwise, the angle between it and your new imaginary line.
Rotate yourself counterclockwise by that same angle, maintaining your distance from the center, like so:
It's okay if you end up making more than a full 360° rotation, just keep on going around the circle until you've doubled the initial angle. For example (assuming the red point is your original position, and the black point is where you end up):
It should be intuitively clear that the further counterclockwise your starting point is from the orienting line, the further you'll travel. In fact, if your starting point is 360° from the orienting line--meaning you start off directly on top of it--doubling your angle will lead you 360° around the circle and right back to where you started.
And if you have a lot of friends doing Step 1 at the same time, it will look something like this:
Step 2 - Distance Adjustment:
Imagine a number line, going from 0 onward:
Take the number line, and imagine placing it on the floor, so that it goes from the center of the room towards (and past) you. The end of the line marked with number 0 should be at the center of the room, and the number 1 should land on the perimeter of the circle we drew. It should look something like this:
Note the number on the number line that corresponds to where you're standing. For instance, if you were standing on the red dot in the above example, your current number value would be something like 1.6 or so. (I totally didn't cheat and find that number by looking at my source code.)
Now, take that number, and square it (a.k.a. multiply that n...
Note: This post assumes you've heard of the Mandelbrot set before, and you want to know more about it, but that you find imaginary and complex numbers (e.g. the square root of negative one) a bit mystifying and counterintuitive. Instead of helping you understand the relevant math like a reasonable person would, I'm just going to pretend the concept doesn't exist, and try to explain how to generate the Mandelbrot set anyway.
My goal is for this post to (theoretically) be acceptable to the historical René Descartes, who coined the term "Imaginary number" because he did not believe such things could possibly exist.
I hereby formally invite you to a dance.
Since we're (presumably) both cool, hip people, let's go with a somewhat avant-garde dance that's popular with the kids these days.
I call this dance the Mandelbrot Waltz, but you can call it whatever you'd like. This dance follows very simple rules, with the quirk that your starting location will influence your part in the dance. You will unfortunately be cursed to dance forever (there's always a catch to these dance invitations!), but if you ever touch the edges of the dance floor, the curse will be lifted and your part in the dance ends, so it's really not all that bad...
In case you don't already know the moves, I'll describe how to do the dance yourself (if given an arbitrary starting point on the dance floor) step-by-step.
How To Perform The Mandelbrot Waltz: A Step-By-Step Guide
Preparation:
You will need: Yourself, an empty room, and a drawing tool (like chalk or tape).
Setup: Draw a line from the center of the room to the nearest part of the wall, like so:
Now, draw a circle around the room's center, such that it intersects the "orienting line" halfway through. It should look something like this:
Starting Position:
Choose a starting point anywhere you want in the room. Remember this position - or jot it down on a notepad if your memory is bad - for later.
Step 1 - Rotation Doubling:
Imagine a line connecting your current position to the center of the circle:
Find the orienting line we drew on the floor earlier, and measure, counterclockwise, the angle between it and your new imaginary line.
Rotate yourself counterclockwise by that same angle, maintaining your distance from the center, like so:
It's okay if you end up making more than a full 360° rotation, just keep on going around the circle until you've doubled the initial angle. For example (assuming the red point is your original position, and the black point is where you end up):
It should be intuitively clear that the further counterclockwise your starting point is from the orienting line, the further you'll travel. In fact, if your starting point is 360° from the orienting line--meaning you start off directly on top of it--doubling your angle will lead you 360° around the circle and right back to where you started.
And if you have a lot of friends doing Step 1 at the same time, it will look something like this:
Step 2 - Distance Adjustment:
Imagine a number line, going from 0 onward:
Take the number line, and imagine placing it on the floor, so that it goes from the center of the room towards (and past) you. The end of the line marked with number 0 should be at the center of the room, and the number 1 should land on the perimeter of the circle we drew. It should look something like this:
Note the number on the number line that corresponds to where you're standing. For instance, if you were standing on the red dot in the above example, your current number value would be something like 1.6 or so. (I totally didn't cheat and find that number by looking at my source code.)
Now, take that number, and square it (a.k.a. multiply that n...