A subjective description of hyperbolic geometry (sans maths).

I have recently posted some of the mathematics associated with hyperbolic geometries. I may, at some future point, add some computational elements, but this time I want to describe hyperbolic geometry without appealing to mathematics at all.

When someone walks away from you, they take up less of your field of view, but they don’t actually shrink. What if they did? Suppose that if someone walked, say, 100 meters away they would shrink to half their original size. To make this work I’m waving my hands vigorously. The halving in size every 100 meters isn’t perfect (it implies pure exponentiation instead of hyperbolic sines), but it’s conceptually the right kind of thing. Walking back to you again they would grow, and from their perspective it’d be you that shrunk, not them, but the shrinking actually happened. This is the case of hyperbolic geometry.

Meaning what? Well, let’s take a for-instance. Suppose someone surrounds you with cars, parked bumper-to-bumper in a big circle 100 meters from you in every direction. It’d take about 125  full-sized sedans cars or 30 tractor-trailers. But if they all shrunk to half their size by virtue of being farther away, it’d take twice that many. This means there would be literally more space on the outside of a ring in hyperbolic geometry than we’d expect. You’ll need a lot more fencing to enclose the same diameter than you would in Euclidean geometry.

This increase in the number of cars needed gets worse the bigger the ring gets. If we make another ring of trucks at 200 meters they’ll shrink by half again, meaning we’ll need 240 instead of the 60 we’d expect. In fact, if we pushed the borders of Wyoming out 100 meters it’d go from a 2042 km border to a 4084 km border; in Euclidean space the extra 100 meters wouldn’t even push it up to 2043 km. And area grows even faster than circumference: then entire area of Wyoming would fit in a circle only a few kilometers across, and the entire USA would only be a about five hundred meters wider.

One way of thinking about this is that hyperbolic geometry is really really roomy. Sticking with out 100 meter rubric, within a half hour’s walk is all of the land in the USA and some to spare. An hour covers more than twice the area of the entire Earth, land and ocean alike.

One downside of this is it is easy to get lost. If I’m off target by a few degrees in Euclidean geometry I’ll still be within sight of my objective after walking for ten minutes. But think how larger the area that can see my objective appears from my starting position. If it’s a kilometer away, in Hyperbolic geometry it looks less than a thousandth as big, meaning I need to be more than a thousand times as accurate to end up anywhere near it. You don’t have to worry about snipers, either: if they’re more than a block away you look so small to them they’ll never hit.

Incidentally, I think this is the ideal geometry for massive online games. No travel need for magical ways to avoid lengthy travel times, plenty of room to add whatever you want to add, and the chance someone will stumble onto something they shouldn’t is very low, since an empty plain is harder to navigate than most Euclidean mazes. So, that’s hyperbolic geometry. Things shrink as they move away, there’s a whole lot of area within a short distance, and it’s very easy to get lost.