Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
You've probably heard the expression "one-dimensional" used to describe someone or something that lacks a certain "depth" of character or complexity. For example, a puppy could be described, more or less, as a creature that only wants to play—sometimes more, sometimes less.
Stereotypes such as the husband who cares only about sports, or the daughter whose only concern is her shoes, offer human examples of this conception of one-dimensionality. One would hope that most people are not so simply described, however.
Taking a broader view of our puppy, we could say that she is also concerned with her hunger level. Given those two primary interests, to describe the puppy at any point in time, we would need two numbers, one representing the desire to play and the other representing the desire to eat.
These two axes serve as the basis for a two-dimensional plane. The points in the plane correspond to different states of our puppy. So, (1, 9), for instance, would represent a puppy that doesn't want to play much but that is extremely hungry. On the other hand, (9,1) would represent a puppy that, perhaps, has just eaten and now is full of energy!
Now that we have the general idea, let's look at a three-dimensional case involving very simple humans. Let's say that these humans have three measurable characteristics: affinity for low-budget movies, truthfulness, and energy level.
Every human can be classified somewhere in this space, depending on one's respective values for the three characteristics. Now, we could ask, "what does it mean for two people to be close to one another in this space?" (Remember that this is not space-space, but rather "characteristic-space.") The best way to think about this is to think about the points corresponding to each person's profile.
Let's say that person A is represented at (1, 9, 9) and person B is represented at (0, 9, 8). This means that person A doesn't like low-budget movies much, is very honest, and has very high energy. Person B can't stand low-budget movies, is very honest, and has high energy. Judging by these characteristics, these two people might get along pretty well. As a rough approximation of their "compatibility," we can find the distance between their profile points in characteristic space by using the 3-D version of the Pythagorean Theorem.
This equals a distance of , or approximately 1.41—very close.
What would we expect of two people who were far apart in this characteristic space? For example, let's consider person C, represented at (1, 0, 0): this person hates low-budget movies, lies like a rug, and spends all day on the couch. Person D, represented at (9, 9, 9), loves low-budget movies, always tells the truth, and works out every day. We can intuitively guess right away that these two probably won't get along; let's see what the distance between them would be:
This expression corresponds to a distance of about 15.5, quite a bit larger than that of the first couple. Of course, in this case, we are looking at only three aspects of a person's life. It's hard to imagine that this would be enough degrees of freedom to come anywhere close to capturing an accurate description of somebody mathematically.
One of the great things about the Internet is its capacity to connect people with the things that they want or need. Many websites collect information about people and then make recommendations as to what book they should read, what music they should listen to, and even whom they should date. Services such as these, however, use many more than just three measurements or dimensions to quantify a person. They typically construct a many-dimensional profile of a person and put it into what is called a "feature vector." This process basically uses information that a person provides to assign that person to a point in a multi-dimensional space.
Let's examine the case of an online dating service. As of this writing, one popular service uses 30 dimensions to quantify a person. The person is then assigned a point in 30-dimensional space. Users then answer questions about their ideal match, thereby creating a virtual 30-dimensional profile. Individuals who are "close" to this person's ideal match profile in 30-dimensional space are considered to be potential romantic matches.
Now, the efficacy of this method could be debated—real humans are not necessarily well-described by only 30 characteristics. Furthermore, not all traits are as important as others; smoking might be a deal breaker, whereas snoring might not be so bad. Nuances such as these are missed by the rough, all-characteristics-are-equal, 30-D distance model. Nonetheless, this system is an example of how many-dimensional objects are at play in our daily lives.
In this example, we used the idea that distance between points is a concept that generalizes no matter what dimension of space we are in. We saw in a previous section that this works for two- and three-dimensional spaces, and we can use the same method to show that it works in four dimensions as well. Of course, we can't empirically verify a distance in four or more dimensions, but the math works. This exemplifies an important idea in mathematics: concepts from spaces or things that we do understand can be expanded to help us grasp spaces and things that we have no hope of experiencing first hand. This boils down to the belief that once we have a good idea, we can "trust the math" in carrying its application to new contexts. This lights the way forward, as we now turn to a completely different, and equivalent, way to think about dimension.