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More writing for the math

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Bradlee Speice
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blog/2024-11-15-playing-with-fire/1-introduction/index.mdx
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@ -6,6 +6,7 @@ authors: [bspeice]
tags: []
---
Wikipedia describes fractal flames fractal flames as:
> a member of the iterated function system class of fractals
@ -40,18 +41,12 @@ 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. Their mathematical foundations come from a paper written by [John E. Hutchinson](https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf),
but reading that paper isn't critical for our purposes. Instead, we'll focus on building a practical understanding
of how they work. The formula for an IFS is short, but will take some time to unpack:
or IFS. The formula for an IFS is short, but will take some time to unpack:
$$
S = \bigcup_{i=0}^{n-1} F_i(S)
$$
TODO: I'm not sure what the intuitive explanation here is. Is the idea that the solution is all points
produced by applying each function to all points in the solution? And the purpose of the chaos game is
that if we find one point in the solution set, we can effectively discover all the other points?
### Solution set
First, $S$. $S$ is the set of points in two dimensions (in math terms, $S \in \mathbb{R}^2$)
@ -91,6 +86,7 @@ F_i(a_i \cdot x + b_i \cdot y + c_i, d_i \cdot x + e_i \cdot y + f_i)
$$
import transformSource from "!!raw-loader!../src/transform"
import CodeBlock from '@theme/CodeBlock'
<CodeBlock language="typescript">{transformSource}</CodeBlock>
@ -135,35 +131,63 @@ export const shiftData = simpleData.map(({x, y}) => toData(applyCoefs(x, y, coef
Fractal flames use more complex functions, but they all start with this structure.
<details>
<summary>If you're interested in more math...</summary>
TODO: Contractive functions, attractors, etc.?
</details>
### Fixed set
With those definitions in place, we can try stating the original problem in
a more natural way:
With those definitions in place, let's revisit the initial problem:
$$
S = \bigcup_{i=0}^{n-1} F_i(S)
$$
> The solution, $S$, is the union of all sets produced by applying each function, $F_i$,
Or, to put it in English, we would get something like this:
> 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 solve the equation, we must already know what the solution is?
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?
TODO: Phrase it another way?
A point is in the solution if it can be reached by applying
one of the functions to another point in the solution?
Is that the definition of a fixed set?
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 wording slightly to match the conventions in the Fractal Flame paper
:::
Before your eyes glaze over, let's unpack this explanation:
- **$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 these points does not change them
- **...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 we care about
Thus, by applying the functions in our system to "fixed points," we will find the other points we care about.
However, this is all a bit vague, so let's work through an example.
<details>
<summary>But if you want a bit more math...</summary>
...then it's worth mentioning some information I've skipped over.
First, the Hutchinson paper explains that the functions $F_i$ must be _contractive_ for this result to hold.
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 $\mathbb{R}^2$ because we're generating images, but the Hutchinson paper
allows for arbitrary dimensions - which means you could also have 3-dimensional fractal flames.
TODO: Mention attractors? https://en.wikipedia.org/wiki/Chaos_game
</details>
## Sierpinski's gasket
With the math out of the way, we're ready to build our first IFS.
The Fractal Flame paper provides us three functions we can use for our system:
The Fractal Flame paper gives us three functions we can use for our function system:
$$
F_0(x, y) = \left({x \over 2}, {y \over 2} \right) \\
@ -175,9 +199,8 @@ $$
### The chaos game
import CodeBlock from '@theme/CodeBlock'
Next, how do we find out all the points in $S$? The paper lays out an algorithm called the "chaos game":
Next, how do we find the "fixed points" we mentioned earlier? The paper lays out an algorithm called the "[chaos game](https://en.wikipedia.org/wiki/Chaos_game)"
that will give us points in the solution set:
$$
\begin{align*}
@ -244,8 +267,9 @@ import chaosGameSource from '!!raw-loader!./chaosGame'
<hr/>
<small>
Note: The image here is slightly different than the one in the paper.
I think the paper has an error, so I'm choosing to plot the image in a way that's consistent with [`flam3` itself](https://github.com/scottdraves/flam3/blob/7fb50c82e90e051f00efcc3123d0e06de26594b2/rect.c#L440-L441).
The image here is slightly different than the one in the paper.
I think the paper has an error, so I'm choosing to plot the image
in a way that's consistent with [`flam3` itself](https://github.com/scottdraves/flam3/blob/7fb50c82e90e051f00efcc3123d0e06de26594b2/rect.c#L440-L441).
</small>
## Weights