You would have to ask an expert (which I am not) to get a better understanding, but to my knowledge there are two general classes of fractals. The first (the type seen below) is a graph of a set of complex numbers. The most well known of these is the Mandelbrot set, whose mathematical equation is:
zn+1 = zn2 + c
where z is a complex number. There’s even a song about it:
The other type of fractal are the pathological monsters (so called because they exhibit what is known in math terms as pathological behavior): Geometric shapes that are self-similar and repetitive. An example is the Koch Snowflake, which is an equilateral triangle with smaller equilateral triangles stuck on each side, over and over infinitely. While building dungeons out of sets of complex numbers is impossible (or at least highly impractical outside of a computer), we can use the rules of these geometric fractals to simply and easily create dungeon maps that appear natural and highly complex.
Here are the simple rules:
- Create a simple shape, a tunnel, a cave, a district, whatever
- make a smaller version and stick them randomly or in an ordered fashion on the outside of or within the original shape (or both!) as appropriate
- repeat until your scale is small enough to stop or you’ve achieved sufficient complexity
And here are some examples. Click through to a larger version of each.
Here is a set of lava tubes, though it could just as easily be a branch of the world tree, an elven tree fort or a glacial fracture.
Here’s a set of caves. This was done by first using fractal technique for the caves themselves, then a second time for the tunnels.
This is a zoom in on one of the caves from above. Inside I’ve put a dwarven city. Some of the districts are in the main cave or some of the nearby smaller caves. Others districts and some paths between them are carved out of the surrounding rock.
That’s all there is to using the repetitive self-similar geometry of fractals to add some natural feel to your maps. Like fractals themselves, a simple rule set can create a high degree of complexity.
If you’d like to see more on fractals outside of gaming applications, here’s a link to a Nova documentary about them. Bring your popcorn though, it’s 53 minutes and you won’t want to pause it:
Interesting approach. While I cannot say that I don’t apply mathematics to my cartographic creations, if I am I don’t apply it consciously. In lots of my maps, however, I don’t apply design fundamentals like varying types of symettery deliberately. Once a map is completed, when I look carefully, I often see assymettrical balance built right in. In an effort to make things look “right”, such design fundementals just seem to invoke themselves – as if done unconsciouly. I give some forethought on the best approach (design wise) to a specific problem or goal. When I create if flows naturally, without lots of preplanning beyond that. Note: I am a professional fantasy cartographer, having created maps on commission for many publishers, so I am coming from an area of expertise.
So I wouldn’t think so deeply as to apply fractal mathematics directly into my maps. I would find that level of preplanning as counter-intuitive.
Back in my college days I went on a kick of designing dungeons and castles following mathematical progressions. These were fun exercises and they were mathematically beautiful but they utterly failed on two important levels:
– First, my players didn’t care. Not only didn’t they care, they didn’t even notice. And when I pointed it out to them (forcing them to notice) they still didn’t care.
– Second, I found these mathematically driven designs not particularly helpful for enlightening things that are important in games such as spaces for habitation and defensible areas for protection.
Since then I’ve adopted an approach that works better for both me and my players. The two principles I follow are “Keep it simple” and step-wise refinement. Meaning, I start with a simple plan and refine it in passes as much as necessary for the story and the group’s interest level.
You just took something VERY complex and made it so easy a caveman could do it.
Wow… and this has been the bane of my “random” map-making existence for DECADES! Thanks!