Curious, right? Worth looking into, eh?
That's not the article you're getting. It started as that article, then I spent 3 hours building a Pythagoras tree fractal. It's 2:30am, and is my life even real?
Who the hell accidentally spends all night building fractals? Me… I guess.
Pretty, innit? Built with React, and it's going to stop working when I npm run build
. Still don't know why. I'll figure that out next week.
Here's how the Pythagoras tree works:
The construction of the Pythagoras tree begins with a square. Upon this square are constructed two squares, each scaled down by a linear factor of ½√2, such that the corners of the squares coincide pairwise. The same procedure is then applied recursively to the two smaller squares, ad infinitum.
That becomes four bullet points:
 1 component called
<Pythagoras >
 draws rectangle
 calculates props for next 2 rectangles
<Pythagoras><Pythagoras>
Which turns into some 30 lines of code:
import React from 'react';
import { interpolateViridis } from 'd3scale';
const Factor = .5*Math.sqrt(2);
const Pythagoras = ({ maxlvl, w, x, y, lvl, left, right }) => {
if (lvl > maxlvl  w < 1) {
return null;
}
const nextLeft = Factor*w,
nextRight = Factor*w,
d = nextLeft + nextRight + w,
A = 45,
B = 45;
let rotate = '';
if (left) {
rotate = `rotate(${A} 0 ${w})`;
}else if (right) {
rotate = `rotate(${B} ${w} ${w})`;
}
return (
<g transform={`translate(${x} ${y}) ${rotate}`}>
<rect width={w} height={w}
x={0} y={0}
style={{fill: interpolateViridis(lvl/maxlvl)}} />
<Pythagoras w={nextLeft}
x={wnextLeft} y={nextLeft} lvl={lvl+1} maxlvl={maxlvl}
right />
<Pythagoras w={nextRight}
x={0} y={nextRight} lvl={lvl+1} maxlvl={maxlvl}
left />
</g>
);
};
export default Pythagoras;
Beautiful. Let me explain.
interpolateViridis
is a d3scale that gives beautiful colors. Call it with an argument in [0, 1]
and it returns a color.
Factor
is the constant linear factor. We use it to calculate the sides of future rectangles.
d
is the diameter of the triangle formed by the current square and two future squares. More on that later.
A
and B
are angles for each future rectangle. Set to 45 degrees statically.
Then We Start Drawing
If we're in a left
rectangle, we set up a left rotation; if right
then a right rotation. rotate()
is an SVG transformation that rotates the current coordinate system.
To draw the rectangle, we:
translate
to(x, y
), that means "move there" add the rotation
 now our coordinate system is moved and rotate
 draw a rectangle at
(0, 0)
 add two
<Pythagoras>
with new parameters
And that's how you build a fractal in React. It won't work in production, but it sure looks pretty on your localhost.
The animation is done in App.js with a timer that updates the maxlvl
prop every 500ms. Calling the root node of Pythagoras
looks like this:

Start lvl
at 0
and set the maxlvl
. Those are important. At maxlvl
past 12 or 13, it stops working. It takes too much CPU power to ever render.
Yes, I tried. The naive algorithm isn't good enough. You could optimize by taking calculations out of recursion and preparing them in advance.
The Part I Can't Figure out
Look at Andrew Hoyer's Pythagoras tree. That thing is beautiful and flexible and dances like treeshaped worm.
I can't figure out how to calculate those angles and rectangle sizes. I know that using .5
in the Factor
is for 45
degree angles.
You can change the ratio by using a .3
and .7
factor for each side. Then it stops working with 45
degree angles yeah.
Ok, that was expected. Since you know all the sides, you should be able to apply the Law of Sines to calculate the angle.
const nextLeft = .3*Factor*w,
nextRight = .7*Factor*w,
d = nextLeft + nextRight + w,
A = Math.degrees(Math.asin(nextRight/d)),
B = Math.degrees(Math.asin(nextLeft/d));
I can't figure it out. I'm pretty sure I'm applying the Law of Sines correctly, but the numbers it throws out are wrong.
Halp!
PS: Here's a paper that describes using Pythagoras trees as data structures. Sort of.
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