What are Dimensions? Vectors, The Fourth-Dimension, and More!
What in the world is a dimension? What even are vectors? Well, read on and maybe your questions may be answered!
So, you want to understand what the fourth-dimension is, and maybe even learn what the fifth-dimension is, huh? Well, it’s quite simple to grasp once we break down things from the beginning. Or at least, as simple as it’ll get.
We first have to start off with the most critical question: what is a dimension, anyways?
As far as “dictionary definition” goes, in physics and mathematics, the dimension of a mathematical space (or object) is informally defined as “the minimum number of coordinates needed to specify any point within it”. So, a surface such as a plane has a dimension of two because two coordinates are needed to specify a point on it. The inside of a cube is considered three-dimensional because three coordinates are needed to locate a point within these spaces.
The most important thing to realise and consider is that dimensions aren’t really physical manifestations; they’re actually mathematical tools that help identify/describe the relative position between objects in space.
To understand complex dimensions, it is crucial to understand the one that we “live in”: a three-dimensional space. A particular diagram may seem vaguely familiar to you if you have any basic knowledge of mathematics and/or physics: the three-dimensional Cartesian coordinate system. Below is a diagram in case you are not familiar with it.
Personally, whenever I see this diagram, I automatically think of vectors. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen (provided that vectors in these directions do not all lie in the same 2-space/plane).In this case, these three values can be labeled by any combination of three chosen from width, height, depth, and breadth.
This all ties back to Euclidean space, which encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. This diagram serves as a three-parameter model of the physical universe (or, at least, the spatial part, without considering time) in which all known matter exists. So, in the “language” of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors, as shown in the diagram (or at least, the diagram alludes to that fact). The cool thing about vectors is that you can pretty much move them around on whatever plane and the values will still remain the same, as long as the directions are still the same. Vectors are also better known as “a quality of phenomenon that has two independent properties: magnitude and direction”.
So, basically, when people say we live in “three-dimensional space”, they actually mean that we live in a world where, if we want to describe the relative position between two objects in space, we require three numbers. Each number indicates how far you need to go in the three possible perpendicular directions in order to get from one object to the other; think of vectors and the three-dimensional Cartesian coordinate system. In math, you may be used to seeing these numbers being assigned variables (x, y, z, etc.), By assigning these variables, you’re describing a point in three-dimensional space. When you assign these three variables a number, you are describing a point in 3-dimensional space corresponding to the so-called origin that you have established. You are basically creating depth, which makes it three-dimensional.
So what is the fourth dimension, then?
A fourth-dimensional space is a space where there is a fourth perpendicular direction in addition to the three perpendicular directions in three-dimensional space. The most common use for fourth-dimensional space in physics is describing the position of events in spacetime, where the fourth direction represents a distance in time. Physicists, when doing calculations involving space time, treat time as if it was another vector in order to make their lives considerably easier. This is why people may refer to time as the “fourth-dimension”, which is true when indulging in physics and mathematics related to spacetime. However, as mentioned before, dimensions are mathematical tools, so they are just abstract concepts designed to result in calculations. However, calling time the “fourth-dimension” outside the context of theoretical calculations related to spacetime is utterly meaningless.
So, let’s start off with a single point. This is simply a one-dimensional construct. Now, extend the point outward, in both directions. You now have a line (yay!). This is a one-dimensional plane, and it includes every possible one-dimensional point. It has infinite length, but no width, height, or depth. I suppose this would be a scalar, considering that it only has magnitude and no defined direction. Take that line and expand it outward in all directions to infinity. We now have a two-dimensional plane, or simply put, a square. It contains every possible line, which can be thought of as having every possible one-dimensional plane within it. It has length and width, but no height, and definitely no depth.
Take the square and extend it upwards and downwards to infinity, and you now possess a three-dimensional plane, which has length, width, and height. This contains every single possible instance of the second-dimension. Now, this is where it gets a bit tricky. Since the fourth dimension is timelike rather than spacelike, we must take our three-dimensional plane and extend it out infinitely backwards and forwards across time. This four-dimensional construct contains every single state of our three-dimensional plane at every possible point in the timeline; past or future. It’s called a world line. Take the worldline and stretch it across every possible progression of time. This new five-dimensional construct holds every possible world line, which means it holds every possible instance of this three-dimensional object in any timeline. You can go on and on, all the way up to the tenth dimension, just by extrapolating outward upon the previous dimension. It’s just building off of what is already there and finding all instances that are possible. In theory, if you could master the fifth and sixth dimension, you could travel back in time or go to different futures. When thinking about dimensions, we can also consider superstrings, the Calabi-Yau manifold, and other morphed facets of what we perceive to be reality.
Well, I’m getting a bit ahead of myself now let’s talk about geometry. So, you had a point, and then made it into a line, and then created a plane, which is a basic square. The three-dimensional equivalent of a square is obviously a cube. So, what is the fourth-dimensional equivalent of a cube? Well, it’s called a tesseract, which is a type of hypercube. A hypercube is a geometric figure in four or more dimensions which is analogous to a cube in three dimensions. Specifically, it is the “n-dimensional” equivalent of a cube for any non-negative integer “n”. Tesseracts move through both time and space without moving in time, which is why they can be so “trippy”. It’s very difficult to imagine a tesseract because, since our brains have evolved in a three-dimensional environment, it’s difficult to imagine things when we don’t have direct experience with them. Consider a 3D cube with a smaller 3D cube within it. Now, imagine as we progress in a new perpendicular direction, the smaller cube will take the place of the larger one; a new small cube appears within the new cube, and so on. Check out this rotating tesseract diagram if you don’t believe me:
Yep, it’s pretty crazy, alright.
Now, just some “food for thought”, but since the concept of “dimensions” are just mathematical tools that help identify the relative position between objects in space, I wouldn’t be surprised that we could use them as vectors and perhaps play with the possibility that black holes can perhaps even rotate. Maybe the universe could have been a result of a four-dimensional star collapsing and forming a black hole, which our universe is contained in? If a black hole truly is an Einstein-Rosen bridge, can it connect us to a different dimension? Can the universe be thought of as just a series of parametric equations within a three-dimensional plane? Who knows.
Maybe one day, I, as well as others, will be able to someday answer these questions to the best of our abilities.