Upward, not Northward.
-- Edwin A. Abbott
Even though I know and will share some tricks to help myself grasp some feelings, I should humbly admit that I don't actually understand the true essence of 4D or higher dimensional objects. I cannot intrinsically construct a mental model for them as any animal can naturally do to 3D stuff, either.
That's exactly part of the fun. It makes hypercubing a "double puzzle": on one hand, it itself represents a collection of complicated combination puzzle, and on the other hand, the mathematical concept behind the puzzles are so confusing but yet intriguing.
My two mental tools to help my high dimensional thinking are analogy and projection. Actually, most videos on youtube you can find, like 1, 2, and 3 apply these two approaches.
If you have read the inspiring novel "Flatland: A romance of many dimensions", then you can skip this section confidently. The book already gave you more than I can do.
We are too accustomed to 3D world so that it is hard to imagine both lower and higher dimensions. The best analogue of a 2D plane is a paper, but the paper is still a 3D object due to its non-zero thickness. It's weird to imagine what is it like to be a 2D being: I can move forward/backward and upward/downward, but not possible to move to
left/right? But still, comparing to trying to understand 4D world, 2D thinking is at least not impossible.
When thinking about 4D space/stuff, it always feels like hitting a wall, and actually worse than that. Normally speaking, before (literally) hitting a wall, you can see it and you can hold yourself back so that you won't hit it, but we don't know how to "see" or "walk" along the 4th direction. All the verbs, terms, and concepts are implicitly defined within our experience in 3D space.
What comes into play here is the power of choosing an analogy to help us do the thought experiment. Obviously, it is difficult to explore the properties of 4D space/stuff as 3D beings (1). Similarly, it should be difficult to even think of the properties of 3D space/stuff as 2D beings. Luckily, we are ourselves 3D beings and are very familiar with
it (2). Also, we can imagine what it is like being as a 2D being (3). With (2) and (3), we should be able to mitigate (1) a little bit, and get some insight about 4D space/stuff. It is also helpful considering 0D->1D->2D->3D->... sometimes.
Consider net as an example. A net of a cube is made up of 6 squares joined by their edges. After being folded, 5 of them goes into the 3rd dimension and all squares enclose the cube. One dimension lower, we can say that a "net" of a square is 4 line segments connected in a row. After being "folded",
3 of them goes into the 2nd dimension and they enclose the square. One dimension higher, not so obvious though, the "net" of a hypercube should be made up of 8 cubes. A convenient way to think of this is that one cube serves as the "bottom" of the to-be-form hypercube, and 6 more cubes join this "bottom" cube with its 6 faces. The rest cube cannot directly attach to the "bottom" cube, because it is the "top" cube that is adjacent to each of the 6 "side" cubes. Once the 8 cubes are "folded", they enclose a hypercube, or mathematical speaking, a tesseract.
Just checkout this video first. It visualize a tesseract quite well. You can get the sense right away as you see the video. Projections are already everywhere through out the history of things, because people want to convey ideas about this world (3D stuff), but useful medium invented so far are 2D-like paper and screen.
We won't be able to touch and feel a real 4D object any time soon, but we can still project 4D object into 3D space, and then reply on computers to draw a 2D projection of the 3D projection of the 4D object on the screen. It is what all those mathematicians and youtubers do to show the idea.
There are many types of projections, but since we are not really doing math research, I would say knowing one of them is useful enough for us: the perspective projection. Check out the 1978 computer animation award video for the application.
This article is more a gateway than an tutorial because many good videos already exist and the visualizations are great. Other than the good materials and my brief hints, one should still try to imagine what are the projection supposed to mean before it was projected so that they can feel more comfortable about all these higher dimensional (yes, not limited to 4D) stuff. Regarding 4D thinking, we are just like the prisoner in the cave, described by Plato in his Utopianism: Without the ability to turning around to look into the truth, we look at projections and imagine what it was like.
介紹了兩種理解高維幾何的心理輔助模型，類比與投影。
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