Thanks very much to reader “ilikemathyoudont” for pointing out that I, yet again, messed up the triangle inequality. Corrected it below. If you are like me and always forget it, you should draw triangles like I do below, and probably you’ll get it right.
I spend a lot of my time (most/all of my time) thinking about shapes/structures/arguments in hyperbolic space, so I thought I’d take a post and explain what it is. Maybe I’ll be able to put up some research-y posts one day about it. NOTE: in this post, we’re only going to talk about two dimensions. In the future we might talk about higher dimensions (like three, the one we live in).
First, let’s intuitively figure out what I mean by “space.” For us, a space is somewhere that a point can walk around and measure how far it’s moved. The way we measure distance is called a metric (I’ve written about metrics before)- here are a few examples. Let’s take as our space the plane a.k.a. a flat land.
The orange path (taxicab metric) takes 14 steps between the two dots. The green only takes 10 (Euclidean metric). I just made up the purple one but it’s a weird length.
Each of these metrics measures a different distance between where a point starts and where it ends. There aren’t too many rules to be a metric:
- you need to have distance zero if and only if start point = end point,
- distances need to be positive (or zero, see 1)
- we need to satisfy the triangle inequality: for any three points x,y,z in your space, this should be true:
I actually use the triangle inequality ALL THE TIME and I always forget what it is and need to draw a triangle. Essentially, you need to be able to draw a triangle with the distances, so the sum of the two sides
can’t must be longer than the third side.
The orange one is fine. The yellow guy on the left isn’t quite a triangle, but his sides are long enough that we can swing around and make a triangle. But poor blue- there’s no way to use those little stumps to close up into a triangle.
So one way you can have a different metric space is by taking your space (like the plane) and putting a different metric on it. But what if we change the space? Like, what if instead of walking around the plane, we walk around a sphere (like the Earth)? Our space will have different properties. For instance, if you walk in a straight line on the plane, you’ll never get back to yourself. But you can walk around the equator (with, yknow, walk-on-water shoes and infinite endurance) and end up right back where you started. Somehow the sphere is fundamentally different from the plane.
Let’s have a super short history lesson. Once upon a time (around 300 B.C.), a Greek named Euclid wrote a super cool book called the Elements. In it he wrote a bunch of definitions and five axioms (things that we assume are true without proof), which laid the groundwork for the study of geometry. These are the axioms:
- A straight line segment can be drawn joining any two points.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are congruent.
- Given a line and a point not on it, there exists exactly one line that passes through that point and never intersects the line.
And here are the pictures that illustrate them:
For a really long time, people were happy with the first four and unhappy with the fifth one. Euclid’s fifth postulate really irked mathematicians for a millenium or so, and in the mid-1800s we finally got non-Euclidean geometry, which accepts the first four postulates and rejects the fifth one.
There are a couple of non-Euclidean geometries (hyperbolic geometry is one of them), but I think spherical geometry is a little bit easier to get your head around first, because the Earth is a sphere (thank you, history). Let’s get back to the metric on the sphere. We want distance to measure the shortest path between any two points, so instead of drawing a straight line on a map we do those funny arcs that they have in airline maps.
Flight map for Air Canada
These arcs look funny, but if you had Venice and Toronto on a globe and taped a piece of floss to the globe between the two, it would map out the arc exactly as it looks on the map above. Turns out all lines in spherical geometry can be extended to great circles on the sphere, a.k.a. the longest possible circle you can draw. The equator is an example of a great circle. Or any circle that includes the north and south poles. And that brings us to Euclid’s postulate no. 5- given a line on the sphere, and a point not on that line, there’s no way to draw a line through the point which is parallel to the first.
No matter how we try to draw a line through the mole, it’ll hit the equator.
Another fun thing about spherical geometry: triangles don’t add up to 180 degrees like they do in Euclidean geometry. This picture from Wikipedia proves it better than I can:
On the sphere, you can have a triangle with way more than 180 degrees: this one has 230 degrees! But in flat space, all triangles add up to 180.
So a few differences we’ve noticed between flat and spherical geometry so far:
- In Euclidean geometry, there’s exactly one line parallel to an original line that goes through some other point. In spherical geometry, there are none.
- In Euclidean geometry, all triangles add up to 180 degrees. In spherical geometry, they add up to more than 180 degrees.
(Side note: this number-of-degrees-in-a-triangle fact is equivalent to the parallel postulate, so these two facts are basically the same).
There’s a natural question that comes up from these two differences:
- Is there a geometry with more than one line parallel to an original that goes through some other point?
- Is there a geometry where all triangles add up to less than 180 degrees?
The answer is yes! This is called hyperbolic geometry and is where lots of research lives nowadays. In this land, if you draw a line there are infinitely many lines parallel to it that go through some other point. And all triangles add up to less than 180 degrees. There are many models of hyperbolic space, but we’ll just look at two. The first one is the Poincare disk model. Escher does a really good picture for this:
All the fish are the same size!
In this model, the outside circle represents the end of space a.k.a. infinity (we call it the circle at infinity or the boundary of hyperbolic space). One way I’ve explained this picture is imagine that there’s an infinitely large bowl printed with all these fish, which are all the same size. If you stick your head into the bowl, the fish at the “bottom” of the bowl will be pretty big, while the fish in your peripheral vision will get smaller and smaller the further away they are from the bottom of the bowl. This is a nice way to start to wrap your head around hyperbolic space, which is fundamentally different from flat space in the opposite way that spherical space is. We say that spheres are positively curved, while hyperbolic space is negatively curved (and flat space isn’t curved or has curvature 0).
The metric is a little harder to see in this model, so mathematicians often use the upper half-space model instead. It’s sort of like using a map to think about the Earth instead of a globe. When we use maps, they’re finite, because the Earth has finite surface area. But in hyperbolic space, since we go off to infinity, we’re going to have to use something that is infinite. So we use the top half of the Euclidean flat grid. While straight lines on a map of Earth are arcs, as we saw above, straight lines on this model of hyperbolic space look a little different.
Straight lines are ones that go straight up to infinity, and segments of half-circles whose diameters lie on the bottom line
This model includes the boundary at infinity too, but it’s infinitely far away up (just like infinity in Euclidean space is infinitely far out). If you have two points (x,y) and (a,b) in the Euclidean plane/flat space, the distance formula (which measures the metric) is . To write this in terms of differentials (nope, not defining that now), we can say for the Euclidean plane. In the upper half plane model of hyperbolic space, the metric is . Roughly, this means that the further up you go, the shorter horizontal distances are. That’s why the fastest way between two points on the bottom line is using those half circles we drew above.
OK that’s our introduction to hyperbolic geometry. I really wanted to put in a math post before my life derails for a bit. So I apologize if we don’t have a post for awhile after this one- I’ll be dealing with a newborn. Here’s a picture of me now: