More Sierpiński’s Triangles inside Pascal’s Triangle? 🤯
Discovering yet more hidden in the seemingly dead zones!
Introduction
I’ve written a couple of articles on Sierpiński’s Triangle in the last couple of days. I thought that was it but yesterday I wondered what would happen if you had more pixel colours, selected over different modulos? I thought modulo 4 should be interesting as it’s a factor of two, so we should get the original structure but with some pattern within the true
regions. Worth a try.
I’m not going to rehash the whole program again; you can copy and paste the complete code from here and see a full explanation of what it’s doing.
Sierpiński modulo 4
I altered the conditional that decides whether a pixel should be black or white to this.
let colour;
switch (modulo) {
case BigInt(0):
colour = 'blue';
break;
case BigInt(2):
colour = 'red';
break;
default:
colour = 'white';
break;
}
point.setAttribute('fill', colour);
The result.
Amazing, I think, but it raises the question.
Can I also populate the remaining dead areas? Not completely.
Working with more colours
I extracted the colour choosing section into a function so that I could rewrite it to produce me an arbitrary number of colours.
function calculateColour(value, maxModulo) {
let colour = (BigInt(360) / BigInt(maxModulo - 1)) * value;
return `hsl(${colour}, 100%, 50%)`;
}
Moving up to eight colours now we get more filling.
With sixty-four colours. We can see much more filling, but there are areas that are uniform.
It seems that as we increase the MAX_MODULO
these areas get fragmented and reduced in size whilst other areas become so complicated that one can no longer see the underlying fractal.
Further Analysis
I want to be able to hide and show selected modulos so that I can see which ones are affecting which regions. To achieve this, I’m going to add the results of each modulo to a g
then create a wee helper function that’ll hide and show those groups which I can call from the browser console.
When I only display the odd numbered modulos I saw something I didn’t expect.
It looks to me like the odd numbered modulos tend to outline the left and right of the regions. It’s not exact, and I admit that I haven’t totally convinced myself that that is indeed what’s happening, however I think it’s a really lovely variation of the fractal.