Math Insight

Examples of n-dimensional vectors

At first, it may seem that going beyond three dimensions is an exercise in pointless mathematical abstraction. One might thing that if we want to describe something in our physical world, certainly three dimensions (or possibly four dimensions if we want to think about time) will be sufficient. It turns out that this assumption is far from the truth. To describe even the simplest objects, we will typically need more than three dimensions. In fact, in many applications of mathematics, it is challenging to develop mathematical models that can realistically describe a physical system and yet keep the number of dimensions from becoming incredibly large.

The following simple examples reveal how quickly the required number of dimensions increases as we try to describe physical objects.

Position of a rigid object

How many dimensions does it take to specify the position of a rigid object (for example, an airplane) in space? Naively, one would think that it would take three dimensions: one each to specify the $x$-coordinate, $y$-coordinate, and the $z$-coordinate of the object. It is correct that one needs only three dimensions to specify, for example, the center of the object. However, even if the center of a rigid object is specified, the object could also rotate. In fact, it can rotate in three different directions, such as the roll, pitch, and yaw of an airplane. Consequently, we need six dimensions to specify the position of a rigid object: three to specify the location of the center of the object, and three to specify the direction in which the object is pointing.

Non-rigid objects

If one considers non-rigid objects, then the number of dimensions required to specify the configuration of the object can be quite high. As a simple example, consider this little guy below.

Stick figure position. It takes a vector in nine-dimensional space to specify the angles of this stick figure's arms, legs, and head. We denote the configuration vector specifying these angles by $\vec{\theta} = (\theta_1, \theta_2, \theta_3, \theta_4, \theta_5, \theta_6,\theta_7, \theta_8,\theta_9)$. You can drag the sliders with your mouse to change the components of the vector. The components $\theta_1$ and $\theta_2$ specify the angles of his left arm, $\theta_3$ and $\theta_4$ specify the angles of his right arm, $\theta_5$, $\theta_6$, $\theta_7$, and $\theta_8$ specify the angles of his left and right legs, and, finally, $\theta_9$ specifies the angle of his head.

More information about applet.

For simplicity, he is only a two-dimensional object as he is confined to lie in the plane of your computer screen. We've fixed his position and the direction his body is pointing. Nonetheless, just to specify the angles of his arms, legs, and head requires a vector in nine-dimensional space. (We'd need even more dimensions if we also wanted to specify his position or his cholesterol level.)

Multidimensional activity of neurons

We run into high dimensional vectors even in fields like neuroscience. Let's say we stick 100 electrodes in the head of our friend Fred, the lab rat, to simultaneously record the activity of 100 of his neurons. Right away, you can see we'll need a 100-dimensional vector to describe Fred's neuronal activity at any point in time. It gets worse, though, if you think about recording Fred's neural activity over an extend period of time. Let's say we record from Fred's neurons while he's working for ten minutes (or 600 seconds). If we sample his neural activity 1000 times a second, that means we will take $1000 \times 600 =$ 600,000 samples during those ten minutes. Multiplying that by the 100 neurons, and we see we need a 60,000,000-dimensional vector to represent Fred's neural activity during those ten minutes.

You can see that a three-dimensional vector isn't large enough for many practical situations.

Is it harder to think about Fred's head than to think about a 60,000,000-dimensional vector? But it is much preferable to put electrodes in Fred's head than in the head of children suffering with severe epilepsy, as still needs to be done in some cases. Won't it be great if we can develop scientific means to avoid the latter?