Fractal Typesetting

Recursive constructions

Fractals are complicated geometric objects that are often described using a recursive procedure. One of the simplest examples is the Sierpinski triangle, which starts with a single equilateral triangle, replaces it with three more equilateral triangles each scaled by the factor \(1/2\), and proceeds recursively from there. You can use the slider below to see this in action:

import {viewof sierp_steps, sierp_approx} from '@mcmcclur/construction-of-the-serpinski-triangle';
viewof sierp_steps

sierp_approx

Typeset fractals??

Now, suppose we start with the symbol \(x\), replace it with \(x_x^x\), and continue recursively. Thus, we might get \[x \to x_x^x \to {x_x^x}_{x_x^x}^{x_x^x} \to {{x_x^x}_{x_x^x}^{x_x^x}}_{{x_x^x}_{x_x^x}^{x_x^x}}^{{x_x^x}_{x_x^x}^{x_x^x}}.\]

I guess you can already start to see what might happen. You can use the slider below to generate the next few levels.

viewof n = Inputs.range([0,7], {step: 1, value: 3})

Other comments

• The animation at the top of the page was produced using the substitution \[x \to {{}_{x}^{x}x_x^x}.\]
• It’s been a while but I believe I learned of this idea from Robert Dickau’s webpage

sierp_pic = typeset_sierpinski("#sierpinski", n)

splash_pic = splash("#splashville");

import {splash, typeset_sierpinski} from './components/typeset_animation.js';