In 1904 the Swedish mathematician Helge von Koch published a paper about a wonderful fractal curve, which since then has variously been called the Koch Curve, Koch Star or Koch Snowflake. I talked about this curve recently on this blog, because I used it to create a Fractal Holiday Decoration on my 3D printer.
Today I wanted to create a way for you to make your own fractal shapes. In the last few days I’ve done a little research on the web around the idea of “making your own fractal”. What I’ve learned is that there are a lot of sites that want to sell you software to make your own fractals, and I was appalled to discover that they’re all pretty much the same.
What these sites all say, in essence, is: “You, the person reading is, are a complete idiot. But there are other people out there somewhere who are a lot smarter than you. They are called ‘mathematicians’, and they do this really arcane, magical mystical thing called ‘mathematics’ — something that you could never hope to understand. But just because you are a hopeless simpleton, well that doesn’t mean you shouldn’t have any fun, or even be allowed outside on occasion. So I’m going to do you a really big favor and let you send me ten bucks so I can give you a way to play around with the Mandelbrot set. You won’t have the faintest idea what’s going on (being a complete idiot and all), but you’ll still be able to make pretty pictures with nice colors.”
I don’t know about you, but there is something about this kind of sales pitch that I find vaguely insulting.
In fact there isn’t anything obscure or mysterious about fractals. Fractals are simply shapes that continue to repeat themselves at progressively smaller scales. If you understand that straightforward concept, then you’re a mathematician. My goal today was to create a sort of fractal playground, so you can make your own fractal shapes — with no mystery at all about what you are doing.
If you’ve read my post about making those holiday decorations, you’ll recall that they were Koch snowflakes, created with a very simple recipe:
- Start with a triangle.
- Glue a 1/3 scale version of the original shape onto the middle of each edge.
- Repeat step (2) with the new smaller edges, until you either get bored or die of exhaustion.
Today I wrote a program to allow you to vary that recipe. Basically, you can create your own starter shape, and see what kind of fractal you get. You can vary how many levels to repeat (the more levels, the crinklier your fractal gets), and what kind of shape to start with (three sided, four sided, etc).
It’s really astonishing how many interesting shapes you can get this way. Try it for yourself by clicking on the image below. Drag the points around to create your own starter shapes, and watch how the very simple recipe above produces a delightful variety of fractals.
Oh, and you don’t need to send me ten bucks. Fractals should be free. 🙂