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Wording tweaks

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Bradlee Speice 2024-12-15 21:35:03 -05:00
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blog/2024-11-15-playing-with-fire/1-introduction/index.mdx
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@ -144,11 +144,11 @@ Or, to put it in English, we might say:
> Our solution, $S$, is the union of all sets produced by applying each function, $F_i$, > Our solution, $S$, is the union of all sets produced by applying each function, $F_i$,
> to points in the solution. > to points in the solution.
There's just one small problem: to find the solution, we must apply these functions to points There's just one small problem: to find the solution, we must already know which points are in the solution.
we know are in the solution. But how do we know which points are in the solution to start with? What?
John E. Hutchinson provides an answer in the [original paper](https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf) John E. Hutchinson provides an explanation 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: defining the mathematics of iterated function systems:
> Furthermore, $S$ is compact and is the closure of the set of fixed points $s_{i_1...i_p}$ > 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 $F$. > of finite compositions $F_{i_1...i_p}$ of members of $F$.
@ -159,7 +159,7 @@ I've tweaked the conventions of that paper slightly to match the Fractal Flame p
Before your eyes glaze over, let's unpack this: 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 - **Furthermore, $S$ 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))**: - **...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 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, - **...of finite compositions $F_{i_1...i_p}$ of members of $F$**: By composing our functions (that is,
@ -188,7 +188,7 @@ This is still a bit vague, so let's work through an example.
## [Sierpinski's gasket](https://www.britannica.com/biography/Waclaw-Sierpinski) ## [Sierpinski's gasket](https://www.britannica.com/biography/Waclaw-Sierpinski)
The Fractal Flame paper gives us three functions to use for our first IFS: The Fractal Flame paper gives three functions to use for a first IFS:
$$ $$
F_0(x, y) = \left({x \over 2}, {y \over 2} \right) \\ F_0(x, y) = \left({x \over 2}, {y \over 2} \right) \\
@ -200,8 +200,8 @@ $$
### The chaos game ### The chaos game
Next, how do we find the "fixed points" we mentioned earlier? The paper lays out an algorithm called the "[chaos game](https://en.wikipedia.org/wiki/Chaos_game)" Next, how do we find the "fixed points" mentioned earlier? The paper lays out an algorithm called the "[chaos game](https://en.wikipedia.org/wiki/Chaos_game)"
that gives us points in the solution set: that gives us points in the solution:
$$ $$
\begin{align*} \begin{align*}
@ -218,7 +218,7 @@ $$
The chaos game algorithm is effectively the "finite compositions of $F_{i_1..i_p}$" mentioned earlier. 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. 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 $[-1, 1]$, 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: so we generate a random point using an existing API: