For the science project I’m currently working on, I’ve been looking a fair bit at Voronoi tiles. These tiles are what you get when you drop a whole bunch of markers onto a plane, and then segment the plane by looking at which marker each point on the plane is closest to. The result usually looks something like this.

However, as the Wikipedia page points out, you don’t have to use the Euclidean distance to compute the tiles. You can use the Manhattan distance instead, if you like. That gives you something like this.

Which is nice. However, I couldn’t help but wondering what shapes you’d get if you used other, perhaps wackier distance metrics. Like, for instance, the Minkowski metric.

Awesome. But why stop there? Why not, for instance, take a look at the sine of the Euclidean distance multiplied by some constant?

Or the cosine, for that matter.

Or the product of the x and y distances…

Or the difference between the Euclidean distance and some designated optimum…

Clearly, there are a whole host of interesting metrics one could use. No doubt someone has explored this already, but it strikes me as a rich vein for visual experimentation.