Things mathematicians know but most people don’t: dimension is malleable
One of my science pet peeves is when people say, “time is the fourth dimension.” It’s like a fishy smell; when someone says it without a hint of humor I usually end up convinced that they don’t know what dimension is or how it works. Considering that nobody really knows how time works either, the statement is just utter nonsense. Even worse are those popular “science” videos with titles like, “Imagining the Tenth Dimension” and, “Who lives in the Eleventh Dimension?”
The biggest misunderstanding is in the word “the.” That is, that time has no choice but to be the fourth dimension. Why can’t time be the first dimension? Or the eleventh? What’s so mystical and supernatural about the number four that time has to go there?
In fact, the reality is that dimension is a mathematical construction designed and redesigned as necessary to suit our needs in analyzing mathematical issues. As a concept, dimension is malleable. And the easiest way to realize this is to notice how many different kinds of dimension there are.
To name a few: the dimension of a vector space, the fractal dimension of a curve, the dimension of a smooth manifold, the Hausdorff & Lebesgue dimensions of a topological space, the Krull dimension of a commutative ring, the Vapnik-Chervonenkis dimension of a hypothesis space (this one is particularly important to me).
These all sound confusing and twisted, and they are almost completely unrelated to each other, and few have anything to do with time or space. So why do we call them all “dimension?” Because their job as concepts is to help us taxonomize mathematical objects with respect to a notion of magnitude or complexity. As a simplified example, we might be able to prove a theorem like: “this computer program can learn to play any two-player game optimally, but only if the ‘game dimension’ is not too large.” Here dimension would describe how complex the game is in a principled way.
On the other side of the coin, “dimensions” often just represent features of some system. For example, you might represent an economic market in terms of the price of tomatoes, diapers, whiskey, corn, and the unemployment rate. Then one instance of the market requires five potentially independent numbers to describe, and hence the space of all markets is a 5-dimensional space. We use the word “space” mathematically to mean “collection.” This has nothing to do with physics.
We can analyze the change of this market over time and, if we assume prices and unemployment rates change continuously, it would trace out a one-dimensional curve through a 5- or 6-dimensional space, depending on how you want to look at things. But is anything about tomatoes or whiskey intrinsic to economics? Certainly not. We could reorder, replace, or remove any of these dimensions if it resulted in a better model. Indeed, the fashion of big data would lead us to believe more dimensions are easier to learn from. The point is that the “true” dimension of an economic market is whatever best suits our analytic needs.
A heavier example is an image: we can call the intensity of each pixel of an image its own dimension, and then the space of all possible 100 by 100 pixel images is a ten-thousand dimensional space! We can ask some very intriguing questions like, what is the dimension of the space of naturally occurring images, or of photographs of cats? Since very few images out of all possible images are actually pictures of cats, we’d expect the dimension to be quite small. In other words, the intensity of some pixels would tend to depend on the intensity of neighboring pixels, and this would reduce the dimension.
People do actually study this question. For example, the mathematical shape and dimension of the space of 3 by 3 patches taken from natural images is known. It’s dimension is 2, and it’s a shape known as the Klein Bottle. The Klein bottle itself is a quirky thing: a two-dimensional surface whose inside is the same as its outside, if you want o interpret it as having four spacial dimensions. Here’s the interpretation in terms of image patches.
Understanding this model leads to improved automated image classification, detection, and generation. But it’s not the only model of natural 3x3 image patches.
Again, the central point here is that we humans invented the notion of dimension and use it as a tool to shape our understanding of various ideas. Spatial or time interpretations of dimension are prevalent because they’re extremely useful, but the truth is that dimension is what we make it. As with the rest of mathematics, it only exists in our minds.
So the next time someone tells you about “the” fourth dimension, tell them that your fourth dimension is corn.
Jeremy Kun is the author of the blog [Math ∩ Programming](jeremykun.com). If you found this article on dimension interesting, you might enjoy an application of the ideas to facial recognition.