Hello, readers. I am back from my blogging break. In today’s post, as I said I would in the last post, I am going to touch upon hyperdimensions. Hyperdimensions are basically dimensions above the dimensions that we experience in everyday life- 1-D, 2-D, 3-D, and the temporal dimension. (Note: Although many people count the dimension of time as the fourth dimension, here in my post, I will just call it the temporal dimension- as many theoretical physicists do- so by fourth dimension I actually mean the fourth spatial dimension, not the time one.) There is no way in which we can see and experience hyperdimensions, given that we are not evolved to do so. So please- don’t be like the foolish amateur trying to construct a 5-d graph- you can’t. However, although we cannot view the hyperdimensions physically, we can view it mathematically and analogically. In this post, I will introduce to you the most basic of the hyperdimensions- the fourth spatial dimension- and have you see it analogically.
The zeroth dimension is a point. The first dimension is an infinite set of points put together in one dimension- say the x-axis dimension- a line. The second dimension is an infinite set of lines put together in a second dimension (the y-axis dimension)- a plane. The third dimension is an infinite set of planes put together in a third dimension (the z-axis dimension)- a cube. If we continue on, the fourth dimension should be an infinite set of cubes put together in the w-dimension. See below.
A note: that picture above of the fourth dimension is WRONG. Why? Because first off, thats basically just two cubes connected with lines by their vertices. Second off, there is no way you can draw a four-dimensional object, because humans can’t visualize it. Humans can’t visualize the w-dimension. I mean, try creating another line (w-axis) that is perpendicular to the x, y, and z axises. You simply can’t, whether on paper or in real life.
Now, before I continue on, what do I mean by “analogically”? Well, it has the word “analogy” in it, so there must be some comparing involved, which is rightly so. By analyzing how a second-dimensional person views the third dimension, we can gain insight on how we, a third-dimensional being, would view the fourth dimension.
So back to the picture. I noted that the 4-d picture was wrong. So what does it actually look like? Again, we can’t see it. But… can we not see a 3-d projection of it? Just think analogically now- suppose you are a 2-d person living on a plane. There is no way you can ever see what a cube actually is, because given you are 2-d, you can only see up to two dimensions. But, however, suppose one day the cube was unfolded and projected onto a 2-d surface, or in other words, we took the net of the cube. (see below left) Then you could see the cube, just not in its 3-d form. Similarly, if we were to “unfold” a 4-d cube, we could perhaps get a 3-d net of it. This 3-d net shown below right is that of a hypercube (4-d cube).
So, looking at the tesseract, if we can somehow fold up the cubes in the fourth dimension, it would form a hypercube- or a fourth-dimensional cube. Of course, can you visualize a possible way to fold up these cubes? No, because once again, we cannot visualize the fourth spatial dimension. We can only see the 3-d form of a 4-d cube.
So far, we have covered possible ways to view the fourth dimension. What is even more interesting, however, is not the dimension itself, but how the fourth and third dimension would interact with each other. Again, we will use analogy. Pretend there is a 2-d person named Flat who is living in a plane. One day, a 3-d guy named Sphere (he’s actually a sphere) decided to visit the plane. Now, remember that Flat can’t view Sphere as an actual sphere because Flat can only see up to two dimensions, just like we can only see up to three spatial dimensions. So what will Flat see if Sphere decides to come into this 2-d world? Well, as I have stated already, “The third dimension is an infinite set of planes…” so Sphere, a 3-d sphere, actually consists of an infinite amount of 2-d circles of different sizes, all stacked up on each other (see left). Thus, when Sphere enters the plane, Flat will only be able to see these 2-d circles that make up the entire sphere. But note again, the circles are not all the same size. For instance, the circles on the uppermost top and bottom of the sphere are small (e.g. the furthest orange slice in the picture) and the circle whose circumference is the equator of the sphere (the middle) is the largest. Now examine the picture at the right. The top picture represents the first scenario, in which Sphere is entering the plane. What will Flat see at that moment? Just a small circle. However, as Sphere is entering the plane more (or pretty much moving down), the circle in which Flat will see gets larger and larger. It will continue to get larger until Sphere is halfway through the plane (the middle picture at right), where the circle Flat sees now is the equatorial circle. Then, as Sphere decides to leave the plane, the last glimpse Flat will see of Sphere is the small circle again (the bottom picture at right). Pretty much, the only way Flat (2-d) can view Sphere (3-d) is the intersections between Sphere and the plane, or in other words, the 2-d components- the different shaped circles- of Sphere.
Again, this is an analogy, so how can we apply this to how we 3-d beings can view 4-d beings if they ever visit our world? Well, just as how Flat viewed Sphere, we are only able to see the 3-d components of the 4-d beings. In other words, we can only view their intersections with our 3-d world. In the case of Sphere and Flat, the intersection came in the form of planes. In our case, the intersection will come in the form of 3-d objects. So if a 4-d guy ever came to our 3-d world, we will be seeing three dimensional blobs. Let me add that these are changing blobs, because look at Sphere- the circles first were small, then big, and small again. The intersections did stay the same throughout did it? No, it changed. Now assume that instead of Sphere, it was you visiting the world. You put your hand through the plane, and what will Flat see? He will see continuously changing 2-d complex shapes as you move your hand through the 2-d plane, because the intersections between your hand and plane will continuously change as you move your hand through the plane. Similarly, we can also expect the same of a 4-d guy- we will see continously changing 3-d blobs, because the intersections between the 4-d guy and our 3-d world not all the same just as the interesections between Sphere/your hand and the plane were not all the same.
I will stop here for now and continue on in my next post, where I will be doing more (more interesting) analogies. Through these analogies in my next post, I will show that if a 4-d person ever visited our world, we would view him as God. Again, I am being very brief on this subject, given that this subject is much more complex than just two or three posts. So I would also advise you to research this on your own. Perhaps one good place to start is Flatland: A Romance of Many Dimensions by Edwin Abbott. Some documentaries on Youtube are also great. I’m afraid that so far I have made this interesting subject appear really boring, but let me tell you that this is a really amazing subject. Hopefully in my next few posts, you will see the true beauty behind hyperdimensions.