by Jason Marshall
There’s been a lot of talk about 2D versus 3D in the television and movie industries lately. But what exactly does one, two, and three-dimensionality really mean—both in the everyday world and in math? Well, prepare to have your curiosity quenched because that’s exactly what we’re talking about today.
But first, the podcast edition of this article was sponsored by Go to Meeting. With this meeting service, you can hold your meetings over the Internet and give presentations, product demos and training sessions right from your PC. For a free, 45 day trial, visit GoToMeeting.com/podcast.
What are Coordinates?
Before we begin talking about what dimensions mean in math, let’s briefly think about a simple but very important question: How do you know where you are? For example, say you’re meeting friends for coffee—what directions do you need to tell them so they can find you?
There are a number of ways to describe your location, but one way is to tell your friend how many blocks from their house they need to walk in the north-south direction (let’s say 5 blocks north), and how many blocks they need to walk in the east-west direction (let’s say 3 blocks east). Additionally, perhaps the building you’re in has two competing coffee shops—one on the first floor and one on the third. If you’re in the third floor coffee shop, you need to give your friend that piece of information too. Believe it or not, by giving these three locations—5 blocks north, 3 blocks east, and the third floor—you’ve just employed a bit of mathematics to define what’s called a coordinate system, and to give your location—your coordinates—in it. Think of it as yet another example that math is quietly ever-present in the everyday world.
What Does “One-Dimensional” Mean?
What does the term “one-dimensional” mean? Well, we know what the “one” part means, but what about the word “dimensional?” The number of dimensions describes the minimum number of directions you can move in and still get to everywhere you could possibly go. For example, something that is one-dimensional only exists along a single direction—in math, this one-dimensional object exists along a line. To see what I mean, imagine again the number line that we first talked about in the article on negative numbers and integers—negative numbers span out to your left, zero is in the middle, and positive numbers go on forever to your right. Notice that this line runs in only one direction—or one dimension—since you only need to move in the direction from your left-to-right (or vise versa) along the line to get to every single possible point on the line. Of course, you can move in the positive or negative “direction” along the line; but don’t be confused: these are not distinct dimensions since both movements occur along the same line. On another note, if you’ve ever wondered where the phrase “he’s so one-dimensional” comes from, the origin and meaning should now be clear—a one-dimensional person doesn’t have a lot of depth (in other words: multiple dimensions) to their character.
What Does “Two-Dimensional” Mean?
So that’s what life in one dimension looks like. How about two? Well, if one-dimensional means you only have to move in one direction to get everywhere you could possibly go, a two-dimensional world must be one where you have to move in a minimum of two directions to get everywhere. Unless you live in a big city with lots of multi-story buildings, you practically live in 2D every day (although don’t take this too literally—we’ll talk in a minute about how you actually live in 3D). As in our example about telling your friend how to meet you for coffee, in order to get anywhere in a city with a grid street pattern, you need to move in two directions (perhaps north-south and east-west if the streets are aligned that way). Mathematically, a two-dimensional object exists in a plane—which you can think of as an infinite sheet of paper. To get to any location on that plane from anywhere else, you need to know two numbers: the distance in the left-right direction from some starting location—called the origin—and the distance in the forward-backward direction from that same origin.
What Does “Three-Dimensional” Mean?
By now the trend should be clear and it probably won’t come as a surprise that to get to every possible place in three-dimensional space, you have to move around in three different directions. As we saw in the coffee shop example, the world we live in is 3D—that’s why we needed three different directions (north-south, east-west, and up-down) to describe the location. Another way that has become very popular in the last decade to describe our location in 3D space on Earth is to use the Global Positioning System (also called GPS). GPS uses a number of satellites to tell you your latitude, longitude, and altitude with very high precision. Once again, GPS uses three numbers (as it must) for the three locations needed to give a position in 3D space. Of course, the fact that we live in 3D space is exactly why 3D TV and 3D movies have become so trendy since, in principle, these technologies have the potential to increase the “realism” of our experiences with these forms of entertainment.
Wrap Up
Okay, that’s all the math we have time for today. Of course, there are many more interesting things to talk about when it comes to coordinate systems and dimensionality in math—like what it means to have four or more dimensions (some theories in physics require ten spatial dimensions!). We’ll come back to this in the future; but next week, in honor of the Math Dude podcast turning the big three-oh (that is, the 30th article), we’re going to premier the first of a special series of “secret agent” math articles—this first one will be on encryption. So be sure to check it out!
Thanks again to our sponsor this week, Go To Meeting. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their online conferencing service.
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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