The Weird Inner World of the 3D Fractal

This post is about one of the really cool things that happens when maths I don’t understand and technologies I don’t understand get together to make something awesome. Let us begin:

A typical fractal, made using the Mandelbrot set.

A fractal is a conceptual object that reveals further details about its shape ad infinitum, upon ever-closer inspection of it’s fabric. Think of the trunk of a tree sprouting branches, which in turn split off into smaller branches, which themselves yield twigs etc and you won’t go far wrong. In fact, fractals typically look like this thing to the right.

These infinitely complex shapes are ‘created’ by instructing graphics software to render the result of a simple mathematics formula. Until now, the result has been a 2D image – there’s no depth, shadow, perspective, or light sourcing. It is a truly abstract mathematical shape.

But since your home computer became powerful enough to do proper image rendering stuff, home hobbyists have begun to innovate on these formulae. For the first time, three dimensional fractals are able to be created with relative ease.

I can’t go into the maths, because you know, I’m not a total geek, but I do want to show you how beautiful some of these shapes are. Let’s run through some examples:

This is what you get by multiplying phi and theta by two.

More like a classic fractal with 0.5*pi to theta and 1*pi to phi.

This time multiplying angle phi by two, but not theta.

But we’re still looking at these things from outside. The really cool bit is when you start to zoom in. So let’s look at some of the high quality renders from the archives of Daniel White at his highly eclectic Skytopia, where I first learned of this phenomenon.

Make sure you click around on some of these thumbnails, yeah?!

If you’re anything like me, you’d be pretty excited at the idea of being able to create both beauty and complexity from something as simple as a few lines of code, and to then be able to explore your creation from every angle.

Then again, if you’re anything like me, you’d feel a bit frustrated that you’ll probably never be able to make something that awesome yourself. So let’s marvel at the wonder of Daniel’s creation as he takes us deep ‘Into the Heart of the Mandelbulb’.

Your comments, please!

Into the Heart of the Mandelbulb

Nature in Numbers

Saw something pretty cool on Boing Boing just now – a short film demonstrating how mathematic principles manifest in nature. It’s something you’ll all have heard about, but the below actually shows you the background, and does so in a really lovely way.

Top marks to filmmaker Cristobal Vila for making Fibonacci, Golden and Angle Ratios, Delaunay Triangulation and Voronoi Tessellations look so darn good.

His website goes further into exploring these ideas:

This section is meant to be a complement to the animation, in order to better understand the theoretical basis that you can find behind the sequences. It was also, more or less, the appearance of the screenplay in the days that I was planning this project.

And goes on to provide great explanations like this:

  • We add a first red seed
  • Turn 137.5º
  • Add a second green color seed and make the previous traveling to the center.
  • Turn other 137.5º
  • Add a third ocher seed and make the previous traveling to the center, to stay side by side with the first one
  • Turn other 137.5º…
  • …and so on, seed after seed, we will obtain gradually a kind of distributions like the ones you have in the following figures

This leads to the characteristic structure in which all seeds are arranged into a sunflower, which is as compact as possible. We have always said: nature is wise 🙂


Web Discoveries for May 27th

These are my links for May 27th