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Archive for the ‘fractals’ Category
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Monday, February 1st, 2010Pentaflakes
Saturday, February 7th, 2009Just a link today: Mike Croucher over at Walking Randomly has some gorgeous pictures of fractal constructions called “pentaflakes”, made by recursively gluing pentagons together in various ways. He’s also made a Mathematica demonstration for playing around with various sorts of “n-flakes” interactively (you don’t need Mathematica to try it out—just the free Mathematica Player).
Carnival of Math #48, and Monday Math Madness #25
Saturday, January 31st, 2009The 48th Carnival of Mathematics is posted at Concrete Nonsense. My favorite posts include Foxmath’s post about a strange iterated sequence involving pi and this amazing picture of a fractal cabbage. Also near and dear to my heart is Mark Dominus’s post on monads and closure operators.
Also, check out Monday Math Madness #25 over at Wild About Math!. This week’s problem is a very interesting counting problem.
Mandelbrot Maps
Monday, September 1st, 2008Run, don’t walk, to play with the Mandelbrot Maps applet, which lets you play around with the Mandelbrot set while seeing the associated Julia sets updated in real time. This fantastic applet was created by Iain Parris, an MSc student of Philip Wadler at the University of Edinburgh. There’s even a short screencast of Iain explaining the math behind the Mandelbrot and Julia fractals and showing some of the cool things you can do with the applet.
Menger sponge video
Saturday, November 10th, 2007Check out the following totally sweet video of zooming into a Menger sponge!
This video was made by David Makin, who has lots of other cool images and videos at his website. You can probably figure out what a Menger sponge is just from watching the video, but it’s a fractal object which is very easy to make. Here’s what you do:
- Start with a solid cube.
- Slice the cube into 27 equal little cubes, by making two parallel slices in each dimension (just like a Rubik’s cube).
- Remove the cube in the very center, and the six cubes in the center of each face of the big cube. You’ll be left with a cube-shaped object with square holes going straight through the middle on each side.
- Repeat this procedure on each of the 20 remaining little cubes, and so on recursively forever.
The Wikipedia page has some nice pictures that should make this pretty clear if it’s not already. In case the video didn’t blow your mind enough, you should note that Menger sponges have zero volume but infinite surface area! (“How is that possible?!” I hear you cry in dismay. Well, infinity plays very weird games with your intuition!)
Found via God Plays Dice.
Fractal Art
Saturday, July 15th, 2006Take a look at the fractal art of Jock Cooper — there are some amazing images there! And tons of them. Here are a couple more or less randomly chosen examples:
And there are hundreds more! He has also made several fractal animations which are very cool (and kind of trippy).
Of course, I’ve written about fractals before. Most of Jock’s images were probably made using a combination of iterated functions of various types (some using complex numbers) and various computer-assisted manipulations.
Thanks to Sam Lawrence for the link. If you ever come across cool math-related things on the web, or have an idea of something you’d like to see me write about, just drop me an e-mail!
The Mandelbrot Set
Monday, May 8th, 2006For those of you already familiar with the Mandelbrot Set, I suppose this will be like visiting with an old friend. For those of you who aren’t — you’re in for a treat!
Okay, you say, that looks pretty cool I guess, but…huh? Well, to answer the fundamental question of “huh?” we need to dust off our Complex Number Skills. (If you don’t know what a complex number is — or need a refresher — read my explanation of complex numbers.)
Here’s what you do. Pick some complex number c and define the function
(Note that z often denotes a complex number.) Now start with 
and iterate the function f, by taking each value output from the function and putting it back into the function. In other words, find 
, and so on. For example, let’s pick 
and follow this process for a few steps:
and so on. This is a very simple process, and can be worked out by hand fairly quickly. It can be worked out by a computer in the blink of an eye.
For some values of c, iterating the function f will tend to produce complex numbers that just get bigger and bigger. For other values of c, iterating the function will produce complex numbers that stay relatively small, no matter how long you keep iterating the function. (Of course there are technical definitions corresponding to the phrases “bigger and bigger” and “stay relatively small”, but for now we won’t worry about what they are.) Try starting with the value 
to see an example of the latter.
Well, now we’re ready to define the Mandelbrot set: the Mandelbrot set is the set of all complex numbers c for which iterating the function produces complex numbers that stay relatively small.
We can make a picture of the Mandelbrot set by letting each complex number 
correspond to the point with coordinates 
. For example, a computer can easily make a picture of the Mandelbrot set by looking at each point on the screen one by one, deciding which complex number c that point corresponds to, then (say) coloring the point black if c is in the Mandelbrot set, and white otherwise. (Often, instead of just white, programs will choose different colors for points which are not in the Mandelbrot set, based on how many iterations the program had to do before it could decide whether the point was in the set or not.)
You might think that with such a simple function, the picture would be simple as well — like a circle, or a parabola, or something like that. But in fact you get that crazy thing shown above. Iteration can make even the simplest functions behave in very complex ways!
In fact, the Mandelbrot set is what is known as a fractal, an object which is infinitely detailed and contains copies (or near-copies) of itself on all different scales. This means that (theoretically) you can keep zooming into the Mandelbrot forever, and you will always see details just as fine and complex as you do at the “top level”. Moreover, as you zoom in, you will find structures that appear to be tiny copies of the entire picture.
But don’t take my word for it — here are some nice zoomed-in pictures of the Mandelbrot set, and you can find lots more with Google image search. You might also want to download some software for viewing fractals to be able to play around with it yourself.
The most amazing thing is that no one made up these pictures — they have existed forever, built into the mathematical structure of the universe, just waiting for someone to come along and iterate a certain function and make a picture out of it. And in fact, it’s only been since the invention of computers that we’ve been able to do such things (although it’s easy to carry out the iteration described above for a particular value of c by hand, to do it for enough different values of c to make a decent picture would take so long, and be so mind-numbingly tedious, as to make it practically impossible.)
More about the Mandelbrot set, if you’re interested:
