I'd like to show you how you can bring some of these incredible shapes into the real world via 3D printing. We'll be using a 3D printing service like Shapeways or Ponoko - so all you'll need is a PC, some free software, a few hours to tinker, and a few bucks for the final 3D print. I suppose this is a little less physical than most Instructables, but these 3D printers are starting to blur the line between real and virtual.
There are 3 main steps to this process, and each involves a different piece of software. You won't need to master any of them, though. We'll just pop in to use the features we need, then pop out and onto the next step. If you're comfortable installing new software and you don't mind poking at a few mysterious menu items then you'll be fine.
Step 1: Gather Your (free) Tools
We'll be using a program called Mandelbulb 3D to generate our shapes. Grab it. This is an app that allows you to explore the strange new world of 3D fractals. M3D is Windows only but it does work on Macs via Wine or CrossOver.
It's a complex, somewhat arcane program - but we'll be concentrating on just a few features. Be warned - if you have a weakness for eye candy or algorithmic art you can easily get lost in the Mandelverse for long stretches of time. ;)
From there we'll move onto an app called Fiji. Grab it. Fiji is cross-platform. I know very little about this one, except for the fact that it will gobble up the stack of images from M3D and spit out the resulting shape as an .OBJ mesh. Easy peasy.
The last stop on the line is Meshlab. Grab it. Meshlab is a powerful app that can perform all manner of topological wizardy. We'll use this to clean up the mesh and prepare it for 3D printing. Opensource and cross-platform.
I've collected these links and a few other resources into a Bitly bundle for easy access.
Get yourself setup with those three apps and let the tinkering commence!