If you’ve ever studied anything related to fractals, you’ve no doubt come across the Mandelbrot set. Named for its progenitor, mathematician Benoit Mandelbrot, the image takes a simple formula, repeats it, studies how quickly or slowly results approach or escape a boundary, and voilà1 — a stunning image is born. And not just at the macro level — zooming in on small pieces of the set reveals striking details. And adding shading to indicate how quickly or slowly numbers escape zero, and you get beautiful images like this:
What’s amazing about that image is that it’s at a magnification of 1 x 1010. If we showed the rest of the Mandelbrot set to scale, it would need to be on a monitor with a diameter of four million kilometers. And yet there’s incredible detail there, all yielded from a simple equation.
The Mandelbrot set has become famous because it’s an elegant and beautiful example of chaos theory in action. Looking at fine detail in the Mandelbrot set gives us insight into, say, how a simple encoding of DNA can cause a tree to grow its branches, or a snail to grow its shell, or a human to grow its brain. Simple, repetitive steps yield results of staggering complexity.
The Mandelbrot set is beautiful and amazing, but it does have a minor flaw: it’s two-dimensional. That’s not a flaw from the standpoint of mathematics. But it does limit its scope — after all, we live in a three-dimensional world, and while we see in the Mandelbrot set tantalizing hints of biology and geology, it is not a representation of our world.
That thought has led Daniel White and Paul Nylander to work together on the “Mandelbulb,” a three-dimensional representation of a Mandelbrot-like set. It is not a literal three-dimensional Mandelbrot set, as that set is generated by an algorithm that has two variables. Essentially, it’s a two-dimensional equation.2
White and Nylander instead worked to create a set like the Mandelbrot set, that would generate a three-dimensional object that showed the same kind of detail at high levels of zoom.
And while they’re still fine-tuning their algorithm, they’ve reached a point at which their project is bearing fruit.
Now, that’s interesting — but the question is what zooming in on the object yields.
The answer is this:
And this:
And this:
And of course, many, many more startling and beautiful and most amazingly, familiar images, images that could well have come from an alien planet or a close-up of coral or an underwater cave.
This universe is written in the language of mathematics. And in many cases, the greatest beauty comes from the simplest equations. For me, this is the thing that makes me feel most connected to, for lack of a better term, a higher power. If there is a God or something similar, It is so much more interesting than a God who simply wills things into being. It’s a creator with subtle, simple, dazzlingly brilliant skill. And if there is no God, if the universe is truly random — well, we have drawn through random chance a very interesting and remarkable universe indeed.
Either way, it’s explorations like this that remind me how very much we’ve learned — and how very much we have yet to learn. And how worth it the journey is.
If you find this sort of thing interesting, you should really take a look at XaoS, a real-time fractal zooming/exploration program. It’s 100% free, runs on Windows, Mac and Unix, and is quite fun to play with.
One of the coolest things about fractals is that they can have non-integer dimensionality. The mandelbrot set has integer dimensionality according to wikipedia, but think about the Sierpinski’s triangle. If you tried to measure the length of all the lines, you’d measure an infinite length (1 dimensional measure). But, because it is all lines, it covers no area (2 dimensional measure). It’s possible to find a dimensionality between 1 and 2 for the fractal which gives a finite measure of it. I don’t remember how to do this, and I’m pretty sure this is not the rigorous way to think about it, but whatever. I like this fact.
Donald Duck in Mathemagic Land
This really does a great job of going into how mathematics is everywhere, in nature. Of course, it’s also funny cause it’s Donald Duck. There are 3 parts, this is the first part, so make sure to watch all of them.
I also remember they released this on one of the Donald Duck Walt Disney Treasure collections, but I forget which one.
Those surfaces brought to my mind electron microscope pictures of viruses and cell surfaces.
Which, properly developed and substantiated, might well be an excellent argument against the creationists who advance the argument that structures such as eyes are too complex to have been developed via evolution and natural selection and that their existence therefore proves the existence of God.
So glad Jackie posted the link to Donald Duck in Mathemagic Land! I loved that when I was a kid, and now I cans how it to my daughter. :)
Indeed, the third and fourth zoom-in pictures immediately reminded me of clathrin-coated pits and vesicles, which are similar to viral bud structures.
In fact, the Mandelbrot set itself is already four-dimensional (well, it has some fractional dimension which may depend on which definition you use, but it sits naturally in a four-dimensional space). What we usually see are two-dimensional slices of it, one of which we call “the Mandelbrot set” and others of which (slices in a different direction) we call “Julia sets”. So you could make a video of successive 3D slices through the Mandelbrot set. Unfortunately, it’s a very wispy tendrilly sort of thing, not easy to visualize, though this video does a credible job. Let me second the recommendation of XaoS as a nice way to interactively explore it.