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(单词翻译:双击或拖选)
by Jason Marshall
There’s been a lot of talk about 2D versus1 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 quenched2 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 Coordinates4?
Before we begin talking about what dimensions mean in math, let’s briefly5 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 coordinate3 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 grid6 street pattern, you need to move in two directions (perhaps north-south and east-west if the streets are aligned7 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 latitude8, longitude9, 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 spatial10 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 premier11 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.
Please email your math questions and comments to........... You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.
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!
收听单词发音 |
1
versus
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| prep.以…为对手，对；与…相比之下 | |
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2
quenched
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| 解（渴）( quench的过去式和过去分词 )； 终止（某事物）； （用水）扑灭（火焰等）； 将（热物体）放入水中急速冷却 | |
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coordinate
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| adj.同等的，协调的；n.同等者；vt.协作，协调 | |
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4
coordinates
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| n.相配之衣物；坐标( coordinate的名词复数 )；（颜色协调的）配套服装；[复数]女套服；同等重要的人（或物）v.使协调，使调和( coordinate的第三人称单数 )；协调；协同；成为同等 | |
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briefly
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| adv.简单地，简短地 | |
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grid
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| n.高压输电线路网；地图坐标方格；格栅 | |
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aligned
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| adj.对齐的，均衡的 | |
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latitude
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| n.纬度，行动或言论的自由(范围)，(pl.)地区 | |
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longitude
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| n.经线，经度 | |
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spatial
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| adj.空间的，占据空间的 | |
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premier
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| adj.首要的；n.总理，首相 | |
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