Speaking of the recent 3D visualization of the Mandelbrot set, here’s a really cool zoom-in on the plain, boring ol’ 2D version (maybe you can put on your Avatar glasses and see the 3D version). At the least, you should break out your bong, Doritos, Pink Floyd, and set up the laser light show, because this thing is seriously trippy.

As a reminder, the Mandelbrot set is an infinitely detailed curve surrounding the region defined by those points which, when iterated through a simple equation (z_n+1 = z_n² + c) remain bounded (the black region); and where the colors in the outside region are determined by the speed at which those points escape to infinity. And when they say that the boundary of this sucker is infinitely long and infinitely complex, they ain’t kidding – this video zooms into an almost indescribably small region. I don’t know that this is the deepest penetration into the Mandelbrot set (probably not), but the scale is so mind-bogglingly huge that it’s impossible to mentally visualize: This quick voyage claims to cover 214 orders of magnitude. Now the ratio of the diameter of a proton to the diameter of the universe is 10⁴². Thus, if you blew a proton up to the size of the universe, and then blew a proton in that structure up to the size of the universe again, you’d have to do this five times to observe a level of detail similar to what is reached in this video. (I don’t know how they programmed this thing to hold numbers of this precision, but I’m impressed.) And at this level of depth, we can see that it is clearly just as complex as it was way back at the beginning.

People get hung up on fractals: How can a 2D curve be infinite in length, yet contained within a finite area? I think this video gives something of a glimpse of how that can be.

I can begin to see why gazing so long into infinity can be so addictive; why people calculate pi out to trillions of digits (as in the last chapter of Contact): by around eight or nine minutes into the video, the radial eightfold symmetry of the journey takes on a particularly mesmerizing quality . . . or maybe I just need to wash out my bong.

The Mandelbrot set is a famous and beautiful fractal that has been known for more than 30 years – anyone who went to college, can identify a Deadhead by smell, or ever owned a black light has probably seen a picture of one, or had a poster on their freshman dorm wall.

But apparently the 3D version has been elusive for decades.

The Mandelbrot set is really just a very crinkly (technical word) boundary on the complex plane between points which satisfy a set of equations (the interior) and points which don’t (the exterior, stretching to infinity). Being a fractal, this boundary is infinitely long (though bounding a finite area), infinitely complex, and infinitely self-similar (though never exactly repeating). The psychedelic coloration usually associated with its dorm posters are generally derived from the speed at which the values of each point diverge from the stable solution required to be within the set.

This structure is, however, ultimately 2D – not in the fractal sense, but in the sense of existing in a plane and not in a space with depth and volume and texture and shading. Apparently, it has been very difficult to find a 3D analog of the infinite complexity of the Mandelbrot set. (3D fractals certainly exist – the Menger sponge, Romanesco broccoli, etc. – but these are repetitively, generically self-similar; not infinitely diverse and non-repeating like the Mandelbrot set.)

This is particularly surprising given the simplicity of the generating function for the Mandelbrot set: Any complex number c, where iteratively finding z_n+1=z_n²+c (starting from z_n=0) doesn’t explode to infinity, is in the set. (Part of the problem may stem from the fact that there is no 3D analog to the complex numbers – only 4D analogs and higher. Still, it intuitively seems that the extra space of 3D should allow something similar to the Mandelbrot set – after all, knots can only be formed in “roomy” spaces of 3D or higher.)

Well, it appears that the 3D equivalent to a Mandelbrot set (or something damned close) has finally been found. And it is amazing and gorgeous.

It’s definitely worth a click through to see all of the incredible images of this thing. (The formula for the beast certainly does appear to be appreciably more hairy, though.) And it looks to have infinite levels of unique but self-similar complexity stretching to the smallest levels:

I, for one, can’t wait to get my black light out.