speice.io/blog/2024-11-15-playing-with-fire/1-introduction.mdx

---
slug: 2024/11/playing-with-fire
title: "Playing with fire: Introduction"
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.

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

import Plot from "react-plotly.js"

<center>
    <Plot
        data={[
            {
                x: [0, 1, 2],
                y: [0, 1, 2],
                type: 'scatter',
                mode: 'markers',
                marker: { size: 15 }
            }
        ]}
        layout={{
            plot_bgcolor: 'rgba(0,0,0,0)',
            paper_bgcolor: 'rgba(0,0,0,0)'
        }}
        config={{
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        }}
    />
</center>

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 $(x + 1, y)$:

<center>
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        data={[
            {
                x: [0, 1, 2],
                y: [0, 1, 2],
                type: 'scatter',
                mode: 'markers',
                marker: { size: 12 },
                name: "(x, y)"
            },
            {
                x: [1, 2, 3],
                y: [0, 1, 2],
                type: 'scatter',
                mode: 'markers',
                marker: { size: 12 },
                name: "(x+1, y)"
            },
            {
                x: [0, 1],
                y: [0, 0],
                mode: 'lines+markers',
                marker: {
                    size: 12,
                    symbol: 'arrow-bar-up',
                    angleref: 'previous',
                    color: 'rgb(0,0,0)'
                },
                type: 'scatter',
                showlegend: false
            },
            {
                x: [1, 2],
                y: [1, 1],
                mode: 'lines+markers',
                marker: {
                    size: 12,
                    symbol: 'arrow-bar-up',
                    angleref: 'previous',
                    color: 'rgb(0,0,0)'
                },
                type: 'scatter',
                showlegend: false
            },
            {
                x: [2, 3],
                y: [2, 2],
                mode: 'lines+markers',
                marker: {
                    size: 12,
                    symbol: 'arrow-bar-up',
                    angleref: 'previous',
                    color: 'rgb(0,0,0)'
                },
                type: 'scatter',
                showlegend: false
            }
        ]}
        layout={{
            plot_bgcolor: 'rgba(0,0,0,0)',
            paper_bgcolor: 'rgba(0,0,0,0)'
        }}
        config={{
            staticPlot: true
        }}
    />
</center>

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

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

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