It’s time to start exploring different geometries! Today we’ll look at taxicab geometry because algebraically, it’s the easiest one to work with. I discussed it briefly before — recall that lines and points are the same as those in the Euclidean geometry we’re used to, but the idea of distance is different. Here, the distance between two points P and Q is how far a taxi would have to drive on a rectangular grid to get from P to Q.
You can see from the figure that the distance between (-2,3) and (3,-1) is 9, since you’d have to drive nine blocks to get from one point to the other. This might seem simple enough, but the situation is quite a bit different from Euclidean geometry.
In Euclidean geometry, the shortest distance between two points is a straight line segment. If you deviate from this segment in any way in getting from one point to the other, your path will get longer.
In taxicab geometry, there is usually no shortest path. If you look at the figure below, you can see two other paths from (-2,3) to (3,-1) which have a length of 9.
Strange! There are a few exceptions to this rule, however — when the segment between the points is parallel to one of the axes. For example, the distance from (0,0) to (4,0) is just 4 — and there’s only one way to get there along a path of length 4. You’ve got to travel along the line segment joining (0,0) and (4,0).
Well, perhaps “strange” is the wrong word. All of this is perfectly normal in taxicab geometry….
Now let’s take a look at triangles. We define them in the usual way — choose three points, and connect them in pairs to form three sides of a triangle. Just like in Euclidean geometry.
Because we’re so familiar with them, I’ve drawn what would be — if we were in the Euclidean realm! — two 3-4-5 right triangles. You’re welcome to verify that OP’Q’ would indeed be a 3-4-5 triangle in Euclidean geometry.
OK, now let’s look at these triangles from the perspective of taxicab geometry. While we can just study these triangles by looking at the graph, it might be helpful to formally define the taxicab distance function. If two points have coordinates and the distance between them is defined to be
First, a word about saying that we’re defining the distance. That’s right — a different distance, like the one we’re used which comes from the Pythagorean theorem, would produce a different geometry. So how your geometry “works” depends upon how you define the distance.
Second, a word about the formula. Take a moment to convince yourself that is how far your taxicab would have to drive in an east-west direction, and is how far your taxicab would have to drive in a north-south direction. Absolute values are needed: the distances and should be positive and equal to each other.
Back to the triangles. In a taxicab world, OPQ is a 3-4-7 triangle! That might strike you as a bit unusual, since in the Euclidean world, the sum of the lengths of any two sides of a triangle is always greater than the length of the third. This is called the triangle inequality, and is usually written
for points X, Y, and Z, where we write to mean the Euclidean distance. In Euclidean geometry, the only way this inequality can actually be an equality is if Y is on the line segment whose endpoints are X and Z — and no other way. But as we’ve just seen,
where P is evidently not on the segment joining O and Q. Yet another difference between taxicab and Euclidean geometries.
Oh, but it gets better…. In the figure above — at least in a Euclidean world — when you rotate two triangles, the angles and lengths remain the same. In other words, the triangles are congruent. It may seem obvious, since clearly the two triangles look the same.
What happens when we “rotate” triangle OPQ in a taxicab world? I put “rotate” in quotes since we haven’t looked at whether or not the Euclidean idea of “angle” has a companion in the taxicab world.
If you do the calculations, you’ll see that OP’Q’ is in fact a 21/5 – 5 – 28/5 triangle! Moreover, even though OQ was the longest side of triangle OPQ, OQ’ is not the longest side of triangle OP’Q’! The longest side is actually OP’.
What’s going on here? We are so used to being able to draw a triangle on a piece of paper — and certainly, if we turn our piece of paper, the lengths of the sides of the triangle don’t change! The triangle stays exactly the same — this is the notion of congruence. But when you change the distance function to the geometry changes. In a very fundamental way, as you see.
Essentially, we need to give up the idea of “angle” in taxicab geometry. That might sound odd — how can you have geometry without angles? As it turns out, there are lots of ways to do this. We’ll be looking at this idea more as this thread continues.
And although there are differences, there are certainly things in common — such as the fact that there is a triangle inequality in both geometries. Is this just coincidence? Mathematically, a distance function is one which satisfies, for points X, Y, and Z:
Now we’ve already encountered properties 2 and 3 so far, and property 1 simply says that two differents points have to be separated by some positive distance.
The concept of a distance function evolved over time — whenever it made sense to talk about distance, it seemed these properties were always in play. And in fact, the distance functions in both Euclidean and taxicab geometry have these three propertes.
So different distance functions produce different geometries. Perhaps the hardest thing about encountering new geometries is putting aside ideas from Euclidean geometry — “forgetting” all about angles, for example, while we’re in a taxicab world. Hopefully, as you learn about more and more non-Euclidean geometries, it will become easier and easier to navigate this Universe of diverse geometrical worlds….