Hyperspace!

When you think of hyperspace, what do you think of? Do you think of your favorite Star Wars movie or a science fiction comic book or do you simply geek out about the fascinating possibility of traveling faster than the speed of light? Hyperspace is defined as higher-dimensional space, meaning space with more than three dimensions. Today, we will focus on the fourth dimension.

We live in a world of three dimensions: length, width and height. Each dimension is perpendicular to the other two. It's our orderly world and it's what we can easily wrap our brains around.

Now imagine there was a fourth dimension, perpendicular to our three. Which way does it extend? Up? Down? Sideways? The answer is surprising. It's invisible to us because it cannot be described in our three dimensions. It's a reality alternate to ours. If you can't visualize it physically, don't worry. Our minds are wired in such a way that that may just be impossible. But I'll walk you through the concepts that allow you to understand where we are coming from.

Let's work our way up to the fourth dimension.

What do zero dimensions look like? It's just a point. No length, no width, no height. No dimensions. Just a point.

What does one dimension look like? It's a line. We only have length. No width, no height. For the purposes of this argument, let's consider a line segment, an object in one dimension, part of the line that represents the full space.

What do two dimensions look like? It's a plane. It's the surface of a piece of paper, extending outward in two directions. We have length and width. No height yet. Let's take that line segment and make it into a new figure in this new direction we now have, at a right angle to the first one. We draw a square.

What do three dimensions look like? This is the space we are used to. Length, width, height. Three directions all perpendicular, all at right angles to each other. We take our square and extend it in our new direction, height. We form a cube.

Let's discuss what we just did. Each time, we took our previous figure and extended it in a new direction, perpendicular to the ones we already had.

So how do we form a four-dimensional cube? We take the cube and extend it into a fourth dimension perpendicular to the other three. A direction that we cannot see or even fathom. This new figure we formed is called a hypercube. Or as popular culture may refer to it, a tesseract. I can't show you the four-dimensional figure itself, but I can show you an animation of its three-dimensional shadow. (Just like the 3D cube forms a 2D shadow, the 4D hypercube forms a 3D shadow.)

As the hypercube moves around, we see this animation of its shadow. It seems impossible, but examine the shadows a cube forms as you move it around and you will note that they seem to morph: from hexagon to triangle to square. It is the same principle with a hypercube.

All this seems a little abstract. How could it be practical to explore something if it doesn't even exist in the real world? If you are of the viewpoint that math only exists as a vehicle for the other sciences, it would seem like this is a waste of time. Why investigate what doesn't exist?

But I'll counter that perspective for two reasons. For one, I truly believe that math is inherently beautiful. There's something innately exhilarating about being able to discuss something that is seemingly so far removed from our day-to-day lives but still be able to know without a doubt certain facts about that construct. That's the universal truth aspect of math. We can extend things that we find out beyond two, three or four dimensions, even to massive numbers of dimensions. It's a fascinating idea to begin to comprehend.

Secondly, the fourth dimension is practical. When we look at a coordinate plane, we are viewing two related ideas, such as time and the height of a plant (at that given time), as a graph in two dimensions. Using three dimensions allows us to view three related variables together. Using four dimensions allows us to view four. And though there is a limit to the number of dimensions we see in physical space, there is not a limit to the number of variables we have to deal with. In fact, in this big-data age we live in, it is very, very useful to keep track of four (or more!) sets of data at a time. In this way, the fourth dimension is applied indirectly in many different ways, and even directly, as we can use the geometry of the fourth dimension (basically just an extension of geometry in 2D and 3D) to relate back to the data involved in various fields of study and discover valuable conclusions.

For this reason, many people consider the fourth dimension to be time. Time is a very important set of data to keep track of. We have three spatial dimensions and we bring in a fourth variable, a fourth set of data, time. This information is often crucial. Say we work in a multiple story building and we want to have a meeting. We can give the coordinates for position (or more practically - the floor, the hall and the room number) but we also need the time to fully determine when the actual meeting occurs. When and where come together so often that many scientists saw it as a valuable and important construct to have the fourth dimension as a time. It may even be considered as not exactly chosen because it is useful, but because together these four dimensions make up a crucial fabric, spacetime. This idea was successfully promoted by Albert Einstein.

The fourth dimension is an interesting concept, and it serves as some food for thought about the line between practical and abstract. There is no right and wrong here necessarily. What the fourth dimension is, and what purposes it should primarily serve is often open to debate. Comment below if you have any questions or would like to share your opinion on the matter.

In this blog post, I laid the foundations for the fourth dimension, so you can begin to understand the concepts behind it and gain the mathematical literacy dealing with this concept. In the next post, I'll discuss some thought experiments and give you some time to think about the abstract and wonderfully perplexing topics of higher dimensions.