Warped Passages - Lisa Randall [9]
The phrase “extra dimensions” is especially baffling because even when we apply those words to space, that space is beyond our sensory experience. Things that are difficult to visualize are generally harder to describe. We’re just not physiologically designed to process more than three dimensions of space. Light, gravity, and all our tools for making observations present a world that appears to contain only three dimensions of space.
Because we don’t directly perceive extra dimensions—even if they exist—some people fear that trying to grasp them will make their head hurt. At least, that’s what a BBC newscaster once said to me during an interview. However, it’s not thinking about extra dimensions but trying to picture them that threatens to be unsettling. Trying to draw a higher-dimensional world inevitably leads to complications.
Thinking about extra dimensions is another thing altogether. We are perfectly capable of considering their existence. And when my colleagues and I use the words “dimensions,” and “extra dimensions,” we have precise ideas in mind. So before taking another step forward or exploring how new ideas fit into our picture of the universe—note the spatial phrases—I will explain the words “dimensions” and “extra dimensions” and what I will mean when I use them later on.
We’ll soon see that when there are more than three dimensions, words (and equations) can be worth a thousand pictures.
What Are Dimensions?
Working with spaces that have many dimensions is actually something everyone does every day, although admittedly most of us don’t think of it that way. But consider all the dimensions that enter into your calculations when you make an important decision, like buying a house. You might consider the size, the schools nearby, the proximity to places of interest, the architecture, the noise level—and the list goes on. You need to optimize in a multidimensional context, enumerating all your desires and needs.
The number of dimensions is the number of quantities you need to know to completely pin down a point in a space. The multidimensional space might be an abstract one, such as the space of features you are looking for in a house, or it might be concrete, like the real physical space we will soon consider. But when buying a house, you can think of the number of dimensions as the number of quantities you would record in each entry in a database—the number of quantities you find worth investigating.
A more frivolous example applies dimensions to people. When you peg someone as one-dimensional, you actually have something rather specific in mind: you mean that the person has only a single interest. For example, Sam, who does nothing but sit at home watching sports, can be described with just one piece of information. If you felt so inclined, you could picture this information as a dot on a one-dimensional graph: Sam’s proclivity to watch sports, for example. In drawing this graph you need to specify your units so that someone else can understand what the distance along this single axis means. Figure 3 shows a plot with Sam as a point along a horizontal axis. This plot represents the number of hours Sam spends per week watching sports on TV. (Fortunately, Sam won’t be insulted by this example; he is not among the multidimensional readers of this book.)
Figure 3. The one-dimensional Sam plot.
Let’s explore this notion a little further. Icarus Rushmore III (Ike in the above story), a Boston resident, is a more complex character. In fact, he is three dimensional. Ike is twenty-one, drives fast cars, and loses money at Wonderland, a town near Boston with a dog-racing track. In Figure 4 I’ve plotted Ike. Although I’ve drawn it on the two-dimensional surface of a piece of paper, the three axes tell us that Ike is definitely three-dimensional. *
Figure 4. The three-dimensional Ike plot. The solid notched lines are the coordinate axes of the three-dimensional plot. The point