---
slug: 2024/11/playing-with-fire
title: "Playing with fire: The fractal flame algorithm"
date: 2024-11-15 12:00:00
authors: [bspeice]
tags: []
---


Wikipedia describes [fractal flames](https://en.wikipedia.org/wiki/Fractal_flame) as:

> a member of the iterated function system class of fractals

It's tedious, but technically correct. I choose to think of them a different way: beauty in mathematics.

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I don't remember exactly when I first learned about fractal flames, but I do remember being entranced by the images they created.
I also remember their unique appeal to my young engineering mind; this was an art form I could participate in.

The original [Fractal Flame Algorithm paper](https://flam3.com/flame_draves.pdf) describing their structure was too much
for me to handle at the time (I was ~12 years old), so I was content to play around and enjoy the pictures.
But the desire to understand it stuck around. Now, with a graduate degree under my belt, I want to revisit it
and try to make some progress.

This guide is my attempt to explain how fractal flames work in a way that younger me &mdash; and others interested in the art &mdash;
can understand without too much prior knowledge.

---

## Iterated function systems

:::note
This post covers section 2 of the Fractal Flame Algorithm paper
:::

As mentioned, fractal flames are a type of "[iterated function system](https://en.wikipedia.org/wiki/Iterated_function_system),"
or IFS. The formula for an IFS is short, but takes some time to unpack:

$$
S = \bigcup_{i=0}^{n-1} F_i(S)
$$

### Solution set

First, $S$. $S$ is the set of points in two dimensions (in math terms, $S \in \mathbb{R}^2$)
that represent a "solution" of some kind to our equation.
Our goal is to find all points in $S$, plot them, and display that image.

For example, if we say $S = \{(0,0), (1, 1), (2, 2)\}$, there are three points to plot:

import {VictoryChart, VictoryTheme, VictoryScatter, VictoryLegend} from "victory";
export const simpleData = [
    {x: 0, y: 0},
    {x: 1, y: 1},
    {x: 2, y: 2}
]

<VictoryChart theme={VictoryTheme.clean}>
    <VictoryScatter data={simpleData} size={5} style={{data: {fill: "blue"}}}/>
</VictoryChart>

However, this is a pretty boring image. With fractal flames, rather than listing individual points,
we use functions to describe which points are part of the solution. This means there are an infinite
number of points, but if we find _enough_ points to plot, we'll end up with a nice picture.
And if the functions change, the solution also changes, and we get a new picture.

### Transform functions

Second, the $F_i(S)$ functions, also known as "transforms."
At their most basic, each $F_i$ takes in a 2-dimensional point and gives back a new point
(in math terms, $F_i \in \mathbb{R}^2 \rightarrow \mathbb{R}^2$).
While you could theoretically use any function, we'll start with a specific kind of function
called an "[affine transformation](https://en.wikipedia.org/wiki/Affine_transformation)."

The general form of an affine transformation is:

$$
F_i(a_i x + b_i y + c_i, d_i x + e_i y + f_i)
$$

import transformSource from "!!raw-loader!../src/transform"
import CodeBlock from '@theme/CodeBlock'

<CodeBlock language="typescript">{transformSource}</CodeBlock>

The parameters ($a_i$, $b_i$, etc.) are values we get to choose.
For example, we can represent a "shift" function like this:

$$
\begin{align*}
a &= 1 \\
b &= 0 \\
c &= 0.5 \\
d &= 0 \\
e &= 1 \\
f &= 1.5 \\
F_{shift}(x, y) &= (1 \cdot x + 0.5, 1 \cdot y + 1.5)
\end{align*}
$$

Applying this function to our original points will give us a new set of points:

import {applyCoefs} from "../src/transform"

export const coefs = {a: 1, b: 0, c: 0.5, d: 0, e: 1, f: 1.5}
export const toData = ([x, y]) => ({x, y})

export const shiftData = simpleData.map(({x, y}) => toData(applyCoefs(x, y, coefs)))

<VictoryChart theme={VictoryTheme.clean}>
    <VictoryScatter data={simpleData} size={5} style={{data: {fill: "blue"}}}/>
    <VictoryScatter data={shiftData} size={5} style={{data: {fill: "orange"}}}/>
    <VictoryLegend
        data={[
            {name: "(x,y)", symbol: {fill: "blue"}},
            {name: "F(x,y)", symbol: {fill: "orange"}}
        ]}
        orientation={"vertical"}
        x={75}
        y={10}
    />
</VictoryChart>

Fractal flames use more complex functions, but they all start with this structure.

### Fixed set

With those definitions in place, let's revisit the initial problem:

$$
S = \bigcup_{i=0}^{n-1} F_i(S)
$$

Or, to put it in English, we might say:

> Our solution, $S$, is the union of all sets produced by applying each function, $F_i$,
> to points in the solution.

There's just one small problem: to find the solution, we must apply these functions to points
we know are in the solution. But how do we know which points are in the solution to start with?

John E. Hutchinson provides an answer in the [original paper](https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf)
explaining the mathematics of iterated function systems:

> Furthermore, $S$ is compact and is the closure of the set of fixed points $s_{i_1...i_p}$
> of finite compositions $F_{i_1...i_p}$ of members of $F$.

:::note
I've tweaked the conventions of that paper slightly to match the Fractal Flame paper
:::

Before your eyes glaze over, let's unpack this:

- **$S$ is [compact](https://en.wikipedia.org/wiki/Compact_space)...**: All points in our solution will be in a finite range
- **...and is the [closure](https://en.wikipedia.org/wiki/Closure_(mathematics)) of the set of [fixed points](https://en.wikipedia.org/wiki/Fixed_point_(mathematics))**:
  Applying our functions to points in the solution will give us other points that are in the solution
- **...of finite compositions $F_{i_1...i_p}$ of members of $F$**: By composing our functions (that is,
  using the output of one function as input to the next function), we will arrive at the points in the solution

Thus, by applying the functions to fixed points of our system, we will find the other points we care about.

<details>
    <summary>If you want a bit more math...</summary>

    ...then there are some extra details I've glossed over so far.

    First, the Hutchinson paper requires that the functions $F_i$ be _contractive_ for the solution set to exist.
    That is, applying the function to a point must bring it closer to other points. However, as the fractal flame
    algorithm demonstrates, we only need functions to be contractive _on average_. At worst, the system will
    degenerate and produce a bad image.

    Second, we're focused on $\mathbb{R}^2$ because we're generating images, but the Hutchinson paper
    allows for arbitrary dimensions; you could also have 3-dimensional fractal flames.

    Finally, there's a close relationship between fractal flames and [attractors](https://en.wikipedia.org/wiki/Attractor).
    Specifically, the fixed points of $S$ act as attractors for the chaos game (explained in more detail below).
</details>

This is still a bit vague, so let's work through an example.

## [Sierpinski's gasket](https://www.britannica.com/biography/Waclaw-Sierpinski)

The Fractal Flame paper gives us three functions to use for our first IFS:

$$
F_0(x, y) = \left({x \over 2}, {y \over 2} \right) \\
~\\
F_1(x, y) = \left({{x + 1} \over 2}, {y \over 2} \right) \\
~\\
F_2(x, y) = \left({x \over 2}, {{y + 1} \over 2} \right)
$$

### The chaos game

Next, how do we find the "fixed points" we mentioned earlier? The paper lays out an algorithm called the "[chaos game](https://en.wikipedia.org/wiki/Chaos_game)"
that gives us points in the solution set:

$$
\begin{align*}
&(x, y) = \text{random point in the bi-unit square} \\
&\text{iterate } \{ \\
&\hspace{1cm} i = \text{random integer from 0 to } n - 1 \\
&\hspace{1cm} (x,y) = F_i(x,y) \\
&\hspace{1cm} \text{plot}(x,y) \text{ if iterations} > 20 \\
\}
\end{align*}
$$

:::note
The chaos game algorithm is effectively the "finite compositions of $F_{i_1..i_p}$" mentioned earlier.
:::

Now, let's turn this into code, one piece at a time.

First, we need to generate some random numbers. The "bi-unit square" is the range $[-1, 1]$,
so we generate a random point using an existing API:

import biunitSource from '!!raw-loader!../src/randomBiUnit'

<CodeBlock language="typescript">{biunitSource}</CodeBlock>

Next, we need to choose a random integer from $0$ to $n - 1$:

import randintSource from '!!raw-loader!../src/randomInteger'

<CodeBlock language="typescript">{randintSource}</CodeBlock>

### Plotting

Finally, implementing the `plot` function. This blog series is designed to be interactive,
so everything displays directly in the browser. As an alternative,
software like `flam3` and Apophysis can "plot" by saving an image to disk.

To show the results, we'll use the [Canvas API](https://developer.mozilla.org/en-US/docs/Web/API/Canvas_API).
This allows us to manipulate individual pixels in an image and display it on screen.

First, we need to convert from Fractal Flame coordinates to pixel coordinates.
To simplify things, we'll assume that we're plotting a square image
with range $[0, 1]$ for both $x$ and $y$:

import cameraSource from "!!raw-loader!./cameraGasket"

<CodeBlock language="typescript">{cameraSource}</CodeBlock>

Next, we'll store the pixel data in an [`ImageData` object](https://developer.mozilla.org/en-US/docs/Web/API/ImageData).
Each pixel in the image on screen has a corresponding index in the `data` array.
To plot a point, we set that pixel to be black:

import plotSource from '!!raw-loader!./plot'

<CodeBlock language="typescript">{plotSource}</CodeBlock>

Putting it all together, we have our first image:

import Playground from '@theme/Playground'
import Scope from './scope'

import chaosGameSource from '!!raw-loader!./chaosGame'

<Playground scope={Scope} noInline={true}>{chaosGameSource}</Playground>

<hr/>

<small>
    The image here is slightly different than in the paper.
    I think the paper has an error, so I'm plotting the image
    like the [reference implementation](https://github.com/scottdraves/flam3/blob/7fb50c82e90e051f00efcc3123d0e06de26594b2/rect.c#L440-L441).
</small>

### Weights

There's one last step before we finish the introduction. So far, each function $F_i$ has
the same chance of being chosen. We can change that by introducing a "weight" ($w_i$)
to each transform in the chaos game:

import randomChoiceSource from '!!raw-loader!../src/randomChoice'

<CodeBlock language={'typescript'}>{randomChoiceSource}</CodeBlock>

import chaosGameWeightedSource from "!!raw-loader!./chaosGameWeighted";

<CodeBlock language={'typescript'}>{chaosGameWeightedSource}</CodeBlock>

For Sierpinski's Gasket, we start with equal weighting,
but changing the transform weights affects how often
the chaos game "visits" parts of the image. If the chaos
were to run forever, we'd get the same image;
but because the iteration count is limited,
some parts may be missing below.

:::tip
Double-click the image if you want to save a copy!
:::

import GasketWeighted from "./GasketWeighted";
import {SquareCanvas} from "../src/Canvas";

<SquareCanvas name={"gasket_weighted"}><GasketWeighted/></SquareCanvas>

## Summary

Studying the foundations of fractal flames is challenging,
but we now have an understanding of both the mathematics
and implementations of iterated function systems.

In the next post, we'll study the first innovation of fractal flames: variations.