space-filling curves

Ok kiddo, imagine you have a piece of paper with a bunch of dots on it. Now, let's say you want to connect all these dots without lifting your pencil off the paper. That's what a space-filling curve is!

But wait, why is it important? Well, sometimes mathematicians need to find a way to cover every single point in a shape, and space-filling curves can do that. It's like coloring in an entire picture without leaving any white spaces.

There are many different types of space-filling curves, but they all work in a similar way. They start at one point and then zig-zag around until they cover every single point in the shape. It might not look like a normal line, but it's still a continuous path that never ends!

One famous example of a space-filling curve is the Hilbert curve. It starts at one corner of a square and then loops around until it fills up the entire square. It's kind of like going on a very long walk, but you always know where you are and you never have to backtrack.

So there you have it, a space-filling curve is like a very special kind of line that covers up every single point in a shape. Isn't math cool?