Transcription
There is a stubbornly persistent myth that the fourth dimension is impossible for us to visualize, depict, or even imagine as anything other than an abstract mathematical notion. This is especially true on YouTube, where math and physics explainers regularly say things along the lines of, "It is impossible for us to visualize four dimensions because we live in a 3D world," or, "Our 3D brains are just incapable of conceptualizing a fourth spatial dimension." This widely held belief is at best severely limiting and at worst completely untrue.
This video, and the series of videos following, are meant to push back against this idea. In this series, we're going to learn how to think in four dimensions. We're going to dissect how we perceive three dimensions and extend these techniques to visualize 4D shapes. We'll learn how 4D geometry works and use geometric logic and lower-dimensional analogies to build up a new world that we can both see on a screen and in our heads. We'll learn to visually understand all sorts of 4D shapes, such as hypercubes, cubers, and hyperspheres, going a lot deeper than you've likely seen before. We'll learn what curved space means, and eventually, for the more mathematical crowd, how to visualize things like quaternions and complex functions in their natural four-dimensional realm.
As this is the introductory video, we'll be covering some of the basics, but even if you have some familiarity with higher dimensions, hopefully, you'll walk away with some new insights. All I want to do in this video is build up an idea of where a fourth dimension lies and what 4D space is. I'm also going to present a simple but versatile mental model of 4D objects that I've never seen anywhere else.
One important point: the question of whether the fourth dimension exists in the physical sense is irrelevant. We will consider four dimensions from a mathematical perspective, not a physical one. But as physics is written in the language of math, thinking four-dimensionally helps with both. Aside from the obvious application of spacetime, there are quaternions, spinners in quantum mechanics, and the strange Hopf vibration that pops up in numerous physical systems. All of these require four dimensions to describe properly.
This series is designed to be, at least up to a point, math-light. That means that the only math you'll need is an understanding of 3D geometry: points, lines, planes, and angles, along with basic notions of length, area, and volume. Being familiar with vectors will certainly help, and though equations of circles and spheres may pop up, we won't be solving very complicated equations. But I make no promise that it will be easy. On the contrary, it can be quite challenging. You'll have to learn new ways of thinking and abandon some of the visual rules that you rely on to navigate a 3D world. It does take a decent amount of motivation. For me, it's become a bit of an obsession. Think of it as an intellectual challenge, but one that is richly rewarding. If you can learn to visualize four dimensions, you'll never see math, physics, or even the universe quite the same way.
Before we begin, let's quickly understand what we mean by dimensions. A single point has zero dimensions, and a line segment has one dimension that we'll call length. We might place this line segment on an axis, let's call it the x-axis. It only takes one number, or coordinate, to specify a location on this axis. A flat shape, such as a square or a circle, lives on a plane and has two dimensions that we might call length and width. The infinite plane it lives on is made from two perpendicular axes: the x-axis and the y-axis. Any point on this plane can be described with a pair of XY coordinates.
Now we can add a third axis called the z-axis, that is at right angles to both the x and y axis, or to the XY plane itself. Now we can talk about solid objects that have volume, like cubes or spheres. It takes three coordinates, XYZ, to describe a location in 3D space. Naturally, a fourth dimension requires four spatial coordinates, which means adding a new axis. We'll often refer to this as the W axis, but the names are arbitrary. This axis has to be perpendicular to the other three, something that is impossible with only three dimensions. In fact, this new axis is perpendicular to what we would think of as our 3D universe. That is what makes it so hard to conceptualize, as we don't have a physical reference for a fourth axis.
Trying to conceptualize the fourth dimension necessarily means trying to imagine something that is beyond our physical universe. It challenges our basic understanding of space and geometry. Questions like, "Where exactly is the fourth dimension?" that can't easily be answered will certainly arise, and we will regularly encounter situations that are simply impossible in 3D. The only way to discuss these ideas without racking our brains is to change our mindset. Instead of trying to force geometric impossibilities into our 3D way of thinking, we simply have to adopt a new way of thinking and work from there. With this mindset, when we encounter something novel that doesn't exist or is impossible in 3D, instead of fighting with our brains, we can simply say that a 4D universe allows such a possibility, and in order to gain a 4D understanding, which we have chosen to undertake, we have to accept it. With enough time, exposure, and reflection, what seemed like an impossibility will eventually feel second nature.
Let's get started. Is the fourth dimension time? This seems like a good place to start, as it's one of the first things people ask when contemplating the fourth dimension. The short answer is that while time can certainly be, and often is, used as a fourth dimension, it is by no means the only way of conceiving a 4D universe. If you limit yourself to physics, time regularly plays the role of the fourth dimension in what we call spacetime. To define an event, you need to know where and when the event occurs. This means you need three spatial coordinates and one temporal, meaning time, coordinate. In Einstein's general relativity, spacetime takes center stage. Mass and energy cause spacetime to curve in on itself, resulting in the physical effect known as gravity. Beyond that, there are more exotic theories in physics that make use of multiple dimensions of space that we can't see.
In this series, however, I won't be focusing on spacetime, general relativity, or physics, at least for now. So let's leave physics itself behind completely for a second. A dimension is an abstract notion that can be almost any type of continuous numerical information. If you want to measure the temperature at every point in a room, you're dealing with four dimensions: three spatial and one, well, thermal. If you conduct a survey that has four numerical questions like age, weight, height, and income level, that's four dimensions of data. In mathematics, the notion of a dimension does not have to correspond to anything physical, but it's helpful to think about them spatially. Basically, you could spend an entire mathematical career thinking about higher dimensions without ever using time.
Our challenge is to learn how to think about a fourth spatial dimension without worrying about what, if anything, it represents. We'll be working in a purely geometric, flat, Euclidean 4D space. Now, that being said, time is a great way to introduce the fourth dimension, as we're so familiar with it. I'll sometimes use it as a crutch or an analogy to a spatial dimension. Think of time as a gateway drug to four-dimensional thinking. For instance, imagine a cube that pops into existence at a certain time, hangs out for 10 seconds, and then vanishes. The cube's existence over that time interval is a four-dimensional object. This is easier to see if we show the cube moving through time along a virtual time axis. Every instant, or slice of time, during that interval represents a three-dimensional cross-section of that cube's existence – three spatial dimensions, like a still frame of a movie. But now consider just the cube's top surface, a two-dimensional square. Over the cube's lifetime, that top square's existence is also three-dimensional: two spatial and one temporal. In fact, every square cross-section of the cube also has its own three-dimensional existence. No matter how you slice it, the four-dimensional object is composed of an infinite number of 3D slices. We'll come back to this idea down the line, as it's crucial to understanding 4D shapes.
But this is also a good time to talk about the limitations of using time as an analogy. For us, time flows in one direction: forwards. We're not able to move backwards in time, and it's not straightforward to think about time as a spatial direction. For instance, how do you measure the angle between a time dimension and a space dimension? How do we rotate an object from a spatial dimension to the time dimension, and wouldn't that process itself take time to occur? Not only do we not need time to visualize four spatial dimensions, but we need time to measure the duration of events. We simply think about space differently than we think about time, and focusing on time exclusively will not help you visualize 4D geometry.
But really, where is the fourth dimension? You can't see it, and you can't touch it, so where the hell is it? The short answer, as I mentioned earlier, is that it is perpendicular to our 3D universe. But what does that even mean? Let's introduce a concept that we will use frequently to help us try to understand higher dimensions: Flatland. If you are at all interested in this topic, you've come across it before. A 2D universe filled with 2D beings that we'll call Flatlanders. Much like us, most Flatlanders have a hard time visualizing higher dimensions. We're going to focus on one particularly intelligent and curious Flatlander named Freddy. Freddy has studied quite a bit of math and understands higher dimensions as a mathematical abstraction, but he desperately wants to visualize a 3D universe, particularly where this strange third dimension is.
Assuming we can communicate with him, we might tell him to imagine a 2D planar universe, just like the one he lives in, but separate from his own. Just as two lines that don't intersect are parallel, this new 2D universe is parallel to his own but just slightly offset. To Freddy, this is a novel concept. Lines can be parallel, but there's only one plane: the plane that he lives in. He's never had to conceptualize another plane before, parallel or otherwise. Even though it's trivial for us, at first, Freddy might imagine these parallel planes as overlapping squares that are offset from each other, like this. The problem with that is that the planes are supposed to be parallel, and therefore they can't intersect. And he can imagine a plane outside his own universe, even if he can't see it directly. This new parallel plane is offset from his own along the third dimension. That means that if you pick any point in Freddy's plane, there is a corresponding point in this parallel plane that is closest to it. The line segment connecting these two points is perpendicular to both planes.
Now, say that there are many parallel planes, each slightly offset from the previous one, and in each plane, we'll mark a point that is closest to the point on the previous plane. Now let's draw a line connecting all the points. This line can be thought of as an axis along the third dimension. Now, all Freddy sees is a single point, but this new axis points through his universe at this point, connecting the corresponding points of all these parallel planes. For the moment, that's the best he can do.
What does this mean for us? In order to think about where the fourth dimension is, we need to think about a three-dimensional space, just like ours, but parallel to our own and slightly offset. What does that mean? There is no parallel space; there's just a space that we live in. But if Freddy can do it, so can we. Just imagine a cube that lives in 3D space, and then imagine another cube that lies in a different 3D space. At this stage, we'll have to rely on our imagination. The two spaces appear to overlap, but somehow they don't intersect, just like two planes can appear to overlap but don't. This is our first exposure to the concept of three-dimensional planes, or hyperplanes. I'll call them three-planes for short.
Now, choose a point in our universe and mark the point closest to it in this parallel three-plane. Now imagine several more hyperplanes, as many as you like, all parallel and offset by some small amount. Each has a point marked that is closest to the point in the plane adjacent to it. Connect all these points to get a line. This line forms an axis that is perpendicular to our home plane, and it represents the fourth dimension. All we see is a single point in our home plane, but this axis points through our universe and extends infinitely in both directions.
Let's try conceptualizing this by falling back on our crutch, time. We'll briefly go back to the idea that there's only one 3D universe, but one that extends forward and backward in time. Now, each parallel plane is just another moment in time, and each point represents a single location in space at a particular time. Connecting all these points gives a timeline corresponding to that particular location in space, and we might think of that timeline as our fourth dimension. But as we have already seen, time only flows one way, and everything that has passed can never be again. All these moments in time will be lost, like tears in rain.
Let's return to our spatial 4D model. Each three-plane exists separately from all the rest, and given the means, we would theoretically be able to travel forwards and backwards through this new dimension, visiting any parallel plane we like, and time is free to be, well, time. But what is this four-dimensional space, exactly? Let's think about Freddy in his flat universe. He's starting to get an idea of where this mysterious third dimension is by thinking about planes outside of and parallel to his universe. He now wants to understand exactly what 3D space is. Let's help Freddy out with an analogy he already knows. You can think of a square as an infinite number of parallel line segments. In fact, you can construct a square by starting off with a line segment and sliding it in a direction perpendicular to itself, generating the square as it moves along. This process is called extrusion.
To understand 3D space, Freddy needs to think about extruding a 2D object. If you take a square and extrude it along a perpendicular direction, you get a 3D rectangular solid, say a cube. This is a new concept for Freddy, but trivial for us. This is like looking at a neat stack of paper. Each sheet of paper is essentially two-dimensional, but with a very thin third dimension: its thickness. Stacking up sheets layer by layer generates a three-dimensional object. In particular, a cube is made up of an infinite number of 2D squares. So Freddy might think of a cube as a hypersquare. This new object takes up space in a 3D universe. We call it volume, but Freddy might think of it as hyperarea. It would take Freddy a while to come around to the idea that from the point of view of the third dimension, solid objects like squares are now thought of as being flat.
Now let's think one dimension higher. To conceptualize 4D space, we have to think about a three-dimensional object like a cube and extrude it along the fourth axis. We're able to do this because, from the point of view of the fourth dimension, three-dimensional objects, which we normally think of as solid, are perfectly flat, simply because they have no extent along the fourth dimension. Just like Freddy, this is hard to accept at first and will definitely take some getting used to. Stacking up 3D cubes like this generates a four-dimensional object, an example of which is a hypercube, also called a tesseract. Each cross-section of the hypercube is three-dimensional and lives in its own 3D hyperplane, just like ours. We can move around in 4D space by moving in a 3D hyperplane and shifting that three-plane back and forth along the fourth axis.
Here it is with all four axes shown. The 4D space this object takes up is referred to as volume and would be measured in quartic units, such as quartic meters, instead of cubic. Now, I will have a lot to say about hypercubes and related shapes in later videos, but we've covered a lot already, and all I want to do for now is to finish up this video by leaving you with an extremely useful mental model of a four-dimensional object to ponder.
As we mentioned, one of the difficult things about visualizing a 4D extrusion is imagining a 3D object as flat. We might be able to imagine it, but it doesn't feel right to us, especially at first. To help us imagine taking a 3D object like a cube, we then project the image of this cube onto a piece of paper or maybe a blank playing card. Now, this is a 2D projection of a 3D object, but we easily associate parts of the drawing with the three dimensions it has: length, width, and depth, lined up with the X, Y, and Z axes. Our brains have no trouble interpreting this as three-dimensional, especially if the image is on a screen and can simulate rotating in three dimensions. We can even plot the position of a particle in 3D space, moving it along the X, Y, and Z directions. Essentially, this card is acting as a virtual three-dimensional plane, but projecting this object onto a flat surface frees up a physical dimension: the dimension that is perpendicular to the card or the screen. We can think of this dimension as our fourth axis, which is perpendicular to the three dimensions depicted on the surface. This allows us to stack up many such virtual hyperplanes, each with a cube projected on it. I call this the deck of cards model. It allows us to conceptualize three dimensions as flat and gives us an easy way to think about a 4D object as the continuum of hyperplanes: the full deck.
Now we can show a particle moving in all four dimensions: the X, Y, and Z directions, and now along the W axis, that is, through the deck. And we can see that motion from multiple angles. If we allow the cards to act as ultra-thin screens, we can show the cubes rotating in virtual 3D space, just like they would on a computer screen. All of this is meant to help us visualize the idea of three-dimensional cross-sections that all stack up to form our four-dimensional hypercube. With the fourth axis pointing through the deck, we'll return to this model frequently, as it gives a great way to visualize the structure of 4D objects in general.
We've explored a lot in this introductory video, establishing a working understanding of the fourth dimension, hyperplanes, and the structure of 4D space. We've used time as an analogy to grasp a fourth spatial dimension and looked at our familiar 3D world from a 2D perspective. We've had a brief glimpse of the hypercube and introduced a powerful tool to visualize the structure of 4D objects. But the most important thing we've done is alter our mindset to see the fourth dimension not as a mystery, but as a natural extension of our world, something that is crucial for embracing and understanding these concepts.
If you've made it this far, thanks for sticking with me. We have a long journey ahead, and we're just getting started. There are many more videos to come, so make sure you don't miss them. Please help this channel grow by hitting the like and subscribe buttons and ringing that notification bell. Thanks for joining me on this journey into the fourth dimension. Safe travels, and I'll see you next time.