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More writing for the main posts

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Bradlee Speice
2024-12-09 22:18:13 -05:00
parent b608a25146
commit 0983558659
16 changed files with 155 additions and 125 deletions
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6
blog/2024-11-15-playing-with-fire/1-introduction/chaosGameWeighted.ts
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@ -3,7 +3,8 @@ import { randomBiUnit } from "../src/randomBiUnit";
import { randomChoice } from "../src/randomChoice";
import { plot } from "./plot"
import {Transform} from "../src/transform";
const iterations = 50_000;
const quality = 0.5;
const step = 1000;
// hidden-end
export type Props = {
@ -20,8 +21,7 @@ export function* chaosGameWeighted(
randomBiUnit()
];
// TODO: Explain quality
const iterations = width * height * 0.5;
const iterations = width * height * quality;
for (let c = 0; c < iterations; c++) {
// highlight-start
const [_, xform] = randomChoice(transforms);

20
blog/2024-11-15-playing-with-fire/1-introduction/index.mdx
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@ -24,7 +24,7 @@ import banner from '../banner.png'
I don't remember exactly when I first learned about 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 participate in.
The original [Fractal Flame](https://flam3.com/flame_draves.pdf) describing their structure was too much
The original [Fractal Flame Algorithm paper](https://flam3.com/flame_draves.pdf) describing their structure was too much
for me to handle at the time (I was ~12 years old), so I was content to play around and enjoy the pictures.
But the desire to understand it stuck around. Now, with a graduate degree under my belt, maybe I can make some progress.
@ -35,7 +35,13 @@ can understand without too much prior knowledge.
## Iterated function systems
As mentioned above, fractal flames are a type of "[iterated function system](https://en.wikipedia.org/wiki/Iterated_function_system),"
:::note
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:
@ -65,7 +71,11 @@ export const simpleData = [
</VictoryChart>
However, this is a pretty boring image. With fractal flames, rather than listing individual points,
we use functions to describe which points are part of the solution. This means there are an infinite
we use functions to describe which points are part of the solution.
TODO: Explain characteristics of the solution - fixed set
This means there are an infinite
number of points, but if we find _enough_ points to plot, we'll end up with a nice picture.
And if we choose different functions to start with, our solution set changes, and we'll end up
with a new picture.
@ -73,10 +83,6 @@ with a new picture.
However, it's not clear which points belong in the solution just by staring at the functions.
We'll need a computer to figure it out.
TODO: Other topics worth covering in this section? Maybe in a `details` block?:
- Fixed sets: https://en.wiktionary.org/wiki/fixed_set
- Compact sets
### 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