[Intro]

Who doesn't like fractals? They're a

beautiful blend of simplicity and

complexity often including these

infinitely repeating patterns.

Programmers in particular tend to be

especially fond of them, because it often takes

a shockingly small amount of code to

produce images that are way more

intricate than any human hand ever could

hope to draw. But a lot of people don't

actually know the definition of a

fractal; at least not the one that Benoit

Mandelbrot, the father fractal geometry,

had in mind. A common misconception is

the fractals are shapes that are perfectly

self-similar. For example this snowflake

looking shape right here called the Von

Koch snowflake, consists of three

different segments, and each one of these

is perfectly self-similar, in that when

you zoom in on it, you get a perfectly

identical copy of the original. Likewise

the famous Sierpinski triangle consists

of three smaller identical copies of

itself. And don't get me wrong, self-similar

shapes are definitely beautiful,

and they're a good toy model for what

fractals really, are but Mandelbrot had a

much broader conception in mind, one

motivated not by beauty, but by a more

pragmatic desire to model nature in a

way that actually captures roughness.

In some ways, fractal geometry is a

rebellion against calculus, whose central

assumption is that things tend to look

smooth if you zoom in far enough.

But Mandelbrot saw this is overly idealized,

or at least needlessly idealized,

resulting in models that neglect the

finer details of the thing that they're

actually modeling, which can matter!

What he observed is that self similar shapes

give a basis for modeling the regularity

in some forms of roughness. But the

popular perception that fractals only

include perfectly self similar shapes is

another over-idealization, one that

ironically goes against the pragmatic

spirit of fractal geometry is origins.

The real definition of fractals has to

do with this idea of fractal dimension

the main topic of this video. You see,

there is a sense, a certain way to define

the word dimension, in which the

Sierpinski triangle is approximately 1.585

dimensional; that the Von Koch curve

is approximately 1.262-dimensional;

the coastline of Britain turns out to be

around 1.21-dimensional.

And in general, it's possible

to have shapes whose dimension is any

positive real number, not just whole

numbers. I think when i first heard

someone reference fractional dimension

like this I just thought it was nonsense,

right? I mean, mathematicians are clearly

just making stuff up. Dimension is

something that usually only makes sense

for natural numbers, right? A line is

one-dimensional; a plane, that's two

dimensional; this space that we live in,

that's three dimensional, and so on.

And in fact, any linear algebra students who

just learned the formal definition of

dimension in that context would agree, it

only makes sense for counting numbers.

And of course the idea of fractal

dimension is just made up.

I mean, this is math, everything is made up.

But the question is whether or not it

turns out to be a useful construct for

modeling the world. And I think you'll

agree, once you learn how fractal

dimension is defined, it's something that

you start seeing almost everywhere that

you look. It actually helps to start the

discussion here by only looking at

perfectly self similar shapes. In fact

I'm going to start with four shapes, the

first three of which aren't even

fractals: A line, a square, a cube, and a

Sierpinski triangle. All of these shapes are

self-similar. A line can be broken up

into two smaller lines, each of which is

a perfect copy of the original just

scaled down by a half. A square can be

broken down into four smaller squares

each of which is a perfect copy of the

original just scaled down by a half.

Likewise, a cube can be broken down into

eight smaller cubes, again

each one is a scaled-down version by

one-half. And the core characteristic of

the Sierpinski triangle is that it's

made of three smaller copies of itself,

and the length of the side of one of

those smaller copies is one-half the

side length of the original triangle.

It's fun to compare how we measure these.

We would say that the smaller line

is one half the length of the original

line, the smaller square is one-quarter

the area of the original square, the

small cube is one eighth the volume of the

original cube and that smaller

sierpinski triangle...

..well we'll talk about how to measure

that in just a moment. What I want is a

word that generalizes the idea of length,

area and volume,

but which I can apply to all of those

shapes and more. Typically in math, the

word that you use for this is "Measure",

but i think it might be more intuitive

to talk about "mass". As in, imagine that

each of these shapes is made out of

metal: A thin wire, a flat sheet, a solid

cube, and some kind of Sierpinski

mesh. Fractal dimension has everything to

do with understanding how the mass of

these shapes changes as you scale them.

The benefit of starting the discussion

with self-similar shapes is that it

gives us a nice clear-cut way to compare

masses. When you scale down that line by

one half, the mass is also scaled down by

one-half, which you can viscerally see

because it takes two copies of that

smaller one to form the whole. When you

scale down a square by one-half, its

mass is scaled down by one fourth, where

again you can see this by piecing

together four of the smaller copies to

get the original. Likewise, when you scale

down that cube by one half, the mass is

scaled down by one eighth, or one-half cubed,

because it takes eight copies of that

smaller cube to rebuild the original.

And when you scale down this Sierpinski

triangle by a factor of a half,

wouldn't you agree that it makes sense

to say that its mass goes down by a

factor of one-third? I mean, it takes

exactly three of those smaller ones to

form the original. But notice that for

the line, the square, and the cube, the

factor by which the mass changed is this

nice clean integer power of one half.

In fact that exponent is the dimension of

each shape. And what's more, you could say

that what it *means* for a shape to be, for example,

two-dimensional, what puts the

"two" in two-dimensional, is that when you

scale it by some factor, its mass is

scaled by that factor raised to the

second power.

And maybe what it means for a shape to

be three-dimensional is that when you

scale it by some factor, the mass is

scaled by the *third* power of that factor.

So if this is our conception of dimension,

what should the dimensionality of the

Sierpinski triangle be? You want to say

that when you scale it down by a factor

of one-half, its mass goes down by

one half to the power of... well whatever its

dimension is. But because it's self-similar,

we know that we want the mass to go

down by a factor of one third. So what's

the number D such that raising one half

to the power of D gives you one third?

Well, that's the same as asking to 2 to the what

equals 3, the quintessential type of

question that logarithms are meant to

answer. And when you go and plug-in

log base two of 3 to a calculator, what

you'll find is that it's about 1.585. so

in this way, this Sierpinski triangle is

not one-dimensional, even though you

could define a curve that passes through

all its points, and nor is it

two-dimensional, even though it lives in

the plane. Instead it's

1.585-dimensional. And if you want to

describe its mass, neither length nor

area seem like the fitting notions. If

you tried its length would turn out to

be infinite, and its area would turn out

to be 0. Instead what you want is

whatever the 1.585-dimensional

analog of length is. Here let's look at

another self-similar fractal, the Von

Koch curve. This one is composed of four

smaller identical copies of itself, each

of which is a copy of the original

scaled down by one third. So the scaling

factor is one-third, and the mass has

gone down by a factor of one-fourth.

That means the dimension should be some

number D so that when we raise one third

to the power of D it gives us one fourth.

Well that's the same as saying 3 to the

what equals four.

So you can go and plug into a calculator

log base 3 of 4, and that comes out to be

around 1.262

so in a sense of on curve is a

1.262-dimensional shape.

Here's another fun one. This is kind of

the right-angled version of the Koch

Curve. It's built up of eight scaled-down

copies of itself, where the scaling

factor here is one fourth.

So if you want to know its dimension it

should be some number D such that one fourth

to the power of D equals one eighth, the

factor by which the mass just decreased.

And in this case the value we want is

log base 4 of 8, and that's exactly three halves.

So evidently, this fractal is

precisely 1.5 dimensional. Does that kind

of makes sense? It's weird, but it's all

just about scaling and comparing masses

while you scale. What I've described

so far, everything up to this point is

what you might call "self-similarity

dimension". It does a good job making the

idea of fractional dimension seem at

least somewhat reasonable, but there's a

problem.

It's not really a general notion. I mean,

when we were reasoning about how the

mass of a shape should change, it relied on

the self similarity of the shapes; that

you could build them up from smaller

copies of themselves. But that seems

unnecessarily restrictive. After all, most

two-dimensional shapes are not at all

self-similar. Consider the disk, the

interior of a circle. We know that it's

two-dimensional, and you can say that this is

because when you scale it up by a factor

of 2, its mass (proportional to the area)

gets scaled by the square of that factor,

in this case four. But it's not like

there's some way to piece together four

copies of that smaller circle to rebuild

the original. So how do we know that the

bigger disk is exactly four times the

mass of the original? Answering that

requires a way to make this idea of mass

a little more mathematically rigorous,

since we're not dealing with physical

objects made of matter, are we? We're

dealing with purely geometric ones

living in an abstract space.

There's a couple ways to think about

this, but here is a common one. Cover the

plane with the grid and highlight all of

the grid squares that are touching the

disk. Now count how many there are. In

the back of our minds, we already know

that a disk is two-dimensional, and the

number of grid squares that it touches

should be proportional to its area. A

clever way to verify this empirically is

to scale up that disk by some factor,

like two, and count how many grid squares

touch this new scaled-up version. You

should find is that that number has

increased approximately in proportion to

the square of our scaling factor, which

in this case means about four times as

many boxes.

Well...admittedly what's on the screen

here might not look that convincing, but

it's just because the grid is really

coarse. If instead you took a much finer

grid, one that more tightly captures the

intent we're going for here by measuring

the size of the circle, that relationship

of quadrupling the number of boxes

touched when you scale the disk by a

factor of two should shine through more

clearly.

I'll admit, though, that when I was

animating this I was surprised by just

how slowly this value converges to 4.

Here's one way to think about this: If

you were to plot the scaling factor

compared to the number of boxes that the

scaled disk touches, your data should very

closely fit a perfect parabola, since the

number of boxes touched is roughly

proportional to the square of the

scaling factor. For larger and larger

scaling values, which is actually

equivalent to just looking at a finer

grid, that data is going to more

perfectly fit that parabola. Now getting

back to fractals, let's play this game

with the Sierpinski triangle, counting

how many boxes are touching points in

that shape. How would you imagine that

number compares to scaling up the

triangle by a factor of 2 and counting

the new number of boxes touched?

Well, the proportion of boxes touched by

the big one to the number of boxes touched

by the small one should be about three.

After all, that bigger version is just

built up of three copies of the smaller

version. You could also think about this

as two raised to the dimension of the

fractal, which we just saw is about 1.585.

And so if you were to go and plot the

scaling factor in this case against the

number of boxes touched by the

Sierpinski triangle, the data would

closely fit a curve with the shape of

y= x^(1.585), just multiplied by some

proportionality constant. But importantly,

the whole reason that i'm talking about

this is that we can play the same game

with non-self-similar shape that still

have some kind of roughness.

The classic example here is the coastline of

Britain. If you plop that coastline into

the plane and count how many boxes are

touching it, then scale it by some

amount and count how many boxes are

touching that new scaled version, what

you'd find is that the number of boxes

touching the coastline increases

approximately in proportion to the

scaling factor raised to the power of

1.21. Here, it's kind of fun to think

about how you would actually compute

that number empirically. As in, imagine I

give you some shape, and you're savvy

programmer. How would you find this

number?

What I'm saying here is that if you

scale this shape by some factor, which

I'll call "s" the number of boxes touching

that shape should equal some constant times

that scaling factor raised to whatever

the dimension is; the value that we're

looking for.

Now if you have some data plot that

closely fits a curve that looks like

the input raised to some power, it can be

hard to see exactly what that power

should be, so a common trick is to take

the logarithm of both sides. That way, the

dimension is going to drop down from the

exponent and we'll have a nice clean

linear relationship. What this suggests

is that if you were to plot the log of

the scaling factor against the log of

the number of boxes touching the

coastline, the relationship should look

like a line, and that line should have a

slope equal to the dimension. So what

that means is that if you tried out a

whole bunch of scaling factors, counted

the number of boxes touching the coast

in each instant, and then plotted the

points on the log-log plot, you could

then do some kind of linear regression

to find the best fit line to your data

set, and when you look at the slope of

that line, that tells you the empirical

measurement for the dimension of what

you're examining. I just think that makes

this idea fractal dimension so much more

real and visceral compared to abstract

artificially perfect shapes. And once

you're comfortable thinking about

dimension like this,

you my friend have become ready to hear

the definition of a fractal. Essentially,

fractals are shapes whose dimension is not an

integer but instead some fractional

amount

What's cool about that is that it's a

quantitative way to say that they're

shapes that are rough, and they stay

rough even as you zoom in.

Technically, there's a slightly more

accurate definition, and I've included it in

the video description, but this idea here

of a non-integer dimension almost

entirely captures the idea of roughness

that were going for.

There is one nuance, though, that I

haven't brought up yet but is worth

pointing out, which is that this

dimension, at least as I've described it so far

using the box counting method, can

sometimes change based on how far

zoomed in you are. For example here's a

shape sitting in three dimensions which

at a distance looks like a line. In 3d, by

the way, when you do a box counting you

have a 3d grid full of little cubes

instead of little squares, but it works

the same way. At the scale where the

shape's thickness is smaller than the

size of the boxes,

it looks one-dimensional, meaning the

number of boxes it touches is

proportional to its length. But when you

scale it up it starts behaving a lot

more like a tube, touching the boxes on

the surface of that tube, and so it'll

look two dimensional, with the number of

boxes touched being proportional to the

square of the scaling factor. But it's

not really a tube, it's made of these

rapidly winding little curves, so once

you scale it up even more, to the point

where the boxes can pick up on the

details of those curves, it looks

one-dimensional again, with the number of

boxes touched scaling directly in

proportion to the scaling constant. So

actually assigning a number to a shape

for its dimension can be tricky and it

leaves room for differing definitions

and different conventions. In a pure math

setting there are indeed numerous

definitions for dimension, but all of

them focus on what the limit of this

dimension is at closer and closer zoom

levels.

You can think of that in terms of the

plot as the limit of this slope as you

move farther and farther to the right.

So for a purely geometric shape to be a

genuine fractal, it has to continue

looking rough even as you zoom in

infinitely far. But in a more applied

setting, like looking at the coastline of

Britain, it doesn't really make sense to

talk about the limit as you zoom in more

and more. I mean at some point you'd just

be hitting atoms. Instead what you do is

you look at a sufficiently wide range of

scales from very zoomed out up to very

zoomed in, and compute the dimension at

each one. And in this more applied

setting, a shape is typically considered

to be a fractal only when the measured

dimension stays approximately constant,

even across multiple different scales.

For example, the coastline of Britain

doesn't just look 1.21-dimensional at

a distance. Even if you zoom in by a

factor of a thousand, the level of

roughness is still around 1.21. That

right there is the sense in which many

shapes from nature actually are self-similar

albeit not perfect

self-similarity. Perfectly self-similar

shapes do play an important role in

fractal geometry. What they give us are

simple-to-describe low-information

examples of this phenomenon of roughness;

roughness that persists at many

different scales and arbitrarily close

scales. And that's important! It gives us

the primitive tools for modeling these

fractal phenomena. But I think it's also

important not to view them as the

prototypical examples of fractals, since

fractals in general actually have a lot

more character to them.

I really do think that this is one of

those ideas where once you learn it, it

makes you start looking at the world

completely differently. What this number

is, what this fractional dimension gives

us, is a quantitative way to describe

roughness. For example the coastline of

Norway is about 1.52-dimensional,

which is a numeric away to communicate

the fact that it's *way* more jaggedy than

Britain's coastline. The surface of a

calm ocean might have a fractal

dimension only barely above two, while a

stormy one might have a dimension closer

to 2.3. In fact, fractal dimension doesn't

just arise frequently in nature, it seems

to be the core differentiator between

objects that arise naturally and those

that are just man-made.

For the final animation here I have a

certain whimsical pi creature fractal I

want to show you. But first I want to

thank the supporters of this channel.

Foremost are those of you contributing on

patreon. Supporters are getting early

access to essence of calculus videos as

I make them, and it's been really nice

having a collection of thoughtful early

viewers to provide feedback. This

particular video was also supported in

part by Affirm. They're this financial tech

company, where I actually used to work,

and these days they're growing a ton.

They're always looking to hire talented

software engineers and data scientists,

and I know that a lot of you out there

watching a video about fractal dimension

just for fun

are that technical talent. Like I said, in

a former life, before I went more into

math, I was actually on their data

science team, and I can tell you the

people there are incredible. I mean, i've

worked with smart teams before, but

Affirm has an unusual concentration of

brilliant minds and technology. I think

it's one of those things where smart

minds tend to attract other smart minds

in this positive feedback loop,

you know what I mean? They're tackling

consumer credit in a pretty novel way

that no one else seems to be doing, and

they can always benefit from a few more

skilled people to help them just do what

they're doing. If you're interested in

applying, I've included a link that's on

the screen and in the description to

their careers page, and this is kind of a

special link, in that if you apply by

first going through that page, it will

let both me and Affirm know that you

heard about them through this video. It's

basically just a way to track the

efficacy of outreach like this, so even

if you don't apply immediately and

instead come back a few days later, it

would help out for clean data's sake if

you still went through that page. So you

should definitely check them out, and

here is that final animation that I

promise you.