---
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](https://en.wikipedia.org/wiki/Fractal_flame) fractal flames as:

> a member of the iterated function system class of fractals

I think of them a different way: beauty in mathematics.

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I don't remember exactly when or how I originally came across fractal flames, but I do remember becoming entranced by the images they created.
I also remember their unique appeal to my young engineering mind; this was an art form I could actively participate in.

The [paper](https://flam3.com/flame_draves.pdf) describing their mathematical structure was too much
for me to handle at the time (I was ~12 years old), and I was content to play around and enjoy the pictures.
But the desire to understand it stuck with me, so I wanted to try again. With a graduate degree in Financial Engineering under my belt,
maybe it would be easier this time.

---

## Iterated function systems

Let's begin by defining an "[iterated function system](https://en.wikipedia.org/wiki/Iterated_function_system)" (IFS).
We'll start at the end and work backwards to build a practical understanding. In mathematical notation, an IFS is:

$$
S = \bigcup_{i=0}^{n-1} F_i(S) \\[0.6cm]
S \in \mathbb{R}^2 \\
F_i(S) \in \mathbb{R}^2 \rightarrow \mathbb{R}^2
$$

### Stationary point

First, $S$. We're generating images, so everything is in two dimensions: $S \in \mathbb{R}^2$. The set $S$ is
all points that are "in the system." To generate our final image, we just plot every point in the system
like a coordinate chart.

TODO: What is a stationary point? How does it relate to the chaos game? Why does the chaos game work?

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>

For fractal flames, we just need to figure out which points are in $S$ and plot them. While there are
technically an infinite number of points, if we find _enough_ points and plot them, we end up with a nice picture.

### Transformation functions

Second, $F_i(S)$. At their most basic, each $F_i$ is a function that takes in a 2-dimensional point and transforms
it into a new 2-dimensional point: $F_i \in \mathbb{R}^2 \rightarrow \mathbb{R}^2$. It's worth discussing
these functions, but not critical, so **this section is optional**.

In mathematical terms, each $F_i$ is a special kind of function called an [affine transformation](https://en.wikipedia.org/wiki/Affine_transformation).
We can think of them like mapping from one coordinate system to another. For example, we can define a coordinate system
where everything is shifted over:

$$
F_{shift}(x, y) = (x + 1, y)
$$

That is, for an input point $(x, y)$, the output point will be $(x + 1, y)$:

export const shiftData = simpleData.map(({x, y}) => { return {x: x + 1, y} })

<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>

This is a simple example designed to illustrate the principle. In general, $F_i$ functions have the form:

$$
F_i(x,y) = (a_i \cdot x + b_i \cdot y + c_i, \hspace{0.2cm} d_i \cdot x + e_i \cdot y + f_i)
$$

The parameters ($a_i$, $b_i$, etc.) are values we get to choose. In the example above, we can represent our shift
function using these parameters:

$$
a_i = 1 \hspace{0.5cm} b_i = 0 \hspace{0.5cm} c_i = 1 \\
d_i = 0 \hspace{0.5cm} e_i = 1 \hspace{0.5cm} f_i = 0 \\
$$

$$
\begin{align*}
F_{shift}(x,y) &= (1 \cdot x + 0 \cdot y + 1, 0 \cdot x + 1 \cdot y + 0) \\
F_{shift}(x,y) &= (x + 1, y)
\end{align*}
$$

Fractal flames use more complex functions to produce a wide variety of images, but all follow this same format.

## Sierpinski's gasket

Using these definitions, we can build the first image. The paper defines a function system we can use as-is:

$$
F_0(x, y) = \left({x \over 2}, {y \over 2} \right)
\hspace{0.8cm}
F_1(x, y) = \left({{x + 1} \over 2}, {y \over 2} \right)
\hspace{0.8cm}
F_2(x, y) = \left({x \over 2}, {{y + 1} \over 2} \right)
$$

### The chaos game

import CodeBlock from '@theme/CodeBlock'

Next, how do we find out all the points in $S$? The paper lays out an algorithm called the "chaos game":

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

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

First, the "bi-unit square" is the range $[-1, 1]$. We can pick a random point like this:

import randomBiunitSource from '!!raw-loader!../src/randomBiunit'

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

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

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

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

Finally, implementing the `plot` function. Web browsers have a [Canvas API](https://developer.mozilla.org/en-US/docs/Web/API/Canvas_API)
we can use for 2D graphics. In our case, the plot function will take an $(x,y)$ coordinate and plot it by
coloring the corresponding pixel in an [ImageData](https://developer.mozilla.org/en-US/docs/Web/API/ImageData):

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

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

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

import Gasket from '!!raw-loader!./Gasket'

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

<hr/>

<small>
Note: The image our chaos game generates is different than the fractal flame paper, but I think the version displayed
here is correct. As confirmation, the next post will re-create the same image using a different method.
</small>

TODO: Explanation of function weights $w_i$