diff --git a/blog/2024-11-15-playing-with-fire/1-introduction/index.mdx b/blog/2024-11-15-playing-with-fire/1-introduction/index.mdx index eeb18f9..c9ecc16 100644 --- a/blog/2024-11-15-playing-with-fire/1-introduction/index.mdx +++ b/blog/2024-11-15-playing-with-fire/1-introduction/index.mdx @@ -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) $$ import transformSource from "!!raw-loader!../src/transform" +import CodeBlock from '@theme/CodeBlock' {transformSource} @@ -135,35 +131,63 @@ export const shiftData = simpleData.map(({x, y}) => toData(applyCoefs(x, y, coef Fractal flames use more complex functions, but they all start with this structure. -
- If you're interested in more math... - - TODO: Contractive functions, attractors, etc.? -
- ### Fixed set -With those definitions in place, we can try stating the original problem in -a more natural way: +With those definitions in place, let's revisit the initial problem: $$ 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. + +
+ But if you want a bit more math... + + ...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 +
## 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'
-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). ## Weights