One of my fellow ScienceBloggers, Andrew Bleiman from Zooilogix, sent me an amusing link. If you've done things like study topology, then you'll know about non-euclidean spaces. Non-euclidean spaces are often very strange, and with the exception of a few simple cases (like the surface of a sphere), getting a handle on just what a non-euclidean space looks like can be extremely difficult.
One of the simple to define but hard to understand examples is called a hyperbolic space. The simplest definition of a hyperbolic space is a space
where if you take open spheres of increasing radius around a point, the amount of space in those open spheres increases exponentially.
If you think of a sheet of paper, if you take a point, and you draw progressively larger circles around the point, the size of the circles increases
with the square of the radius: for a circle with radius R, the amount of space inside the circle is proportional to R2. If you did it in three dimensions, the amount of space in the spheres would be proportional to R3. But it's always a fixed exponent.
In a hyperbolic space, you've got a constant N, which defines the "dimensionality" of the space - and the open spheres around it enclose a
quantity of space proportional to NR. The larger the open circle around
a point, the higher the exponent.
What Andrew sent me is a link about how you can create models of hyperbolic
spaces using simple crochet. And then you can get a sense of just how a hyperbolic space works by playing with the thing you crocheted!
It's absolutely brilliant. Once you see it, it's totally obvious
that this is a great model of a hyperbolic space, and just about anyone
can make it, and then experiment with it to get an actual tactile sense
of how it works!
It just happens that right near where I live, there's a great yarn shop whose owners my wife and I have become friends with. So if you're interested in trying this out, you should go to their shop, Flying Fingers, and buy yourself some yarn and crochet hooks, and crochet yourself some hyperbolic surfaces! And tell Elise and Kevin that I sent you!