I'd like to provide some thought experiments involving the Fourth Dimension that help to reveal just how powerful this concept is.
This first passage comes from Chapter 10 of one of my favorite books, Things to Make and Do in the Fourth Dimension by Matt Parker, an Australian recreational mathematician and popularizer of math. He calls himself a "standup mathematician" as he likes to mash comedy and math together in fun and engaging performances.
The passage concerns what would happen if a 4D creature was to meet our 3D world. It's an entirely fantastical idea, but it's a really neat picture of how the fourth dimension functions.
We are 3D creatures, and as a 3D creature I find no concept as terrifying as the notion of encountering a 4D creature. Such an organism would be god-like to us and, were it the slighest bit malicious, it could torment and destroy us at will. Humans are not equipped physically or mentally to deal with a fourth dimension, so any higher-dimensional being would have the ultimate tactical advantage.
We can explain the disjointed nature of the 4D monster by looking at what would happen if we, as 3D creatures, attacked a 2D creature [...] Let's imagine a hypothetical creature who is completely flat - a hypoflatical, say - living in a completely thin universe, so thin that it would appear as a piece of paper does to us. We could loom as close to it in an up or down direction as we want and, because the hypoflatical has no concept of a third dimension, it would have no idea we were there. The third dimension provides perfect cover. Time to mount our own terrifying attack on a lesser-dimensional being - and all it takes is a move in the third direction into its two-dimensional world.
Parker then begins to describe what would happen if a hand was pressed fingers first into the 2D surface.
Imagine it from the hypoflatical's point of view. As our fingertips pierce its 2D reality, they appear as individual floating circles which grow, move and then merge together as our palm reaches the 2D world. If we pass our fingers through their flat universe, all the hypoflatical sees are shape-shifting 2D cross sections ... Trying to hide won't do it any good either: as 3D creatures, we can see the entire 2D world set out in front of us like a blueprint.
So the hypoflatical cannot take cover and lock itself away safely. We can reach into any closed 2D room as easily as we can reach into the middle of a 2D square [...] This is what makes the 4D attacker so scary to us 3Ders: it can sit right next to our 3D universe watching everything we do and everything inside us, and needs only to reach in and kill us from the inside. They're ominous and gruesome but, thankfully, there's no evidence that 4D creatures exist.
In the same way that our hand created floating cross sections in the 2D world, a 4D creature would create 3D cross sections as it passed through our world. If you were to pass a cube through a 2D world you would create square, rectangular, triangular or hexagonal cross sections depending on the angle in which the cube was rotated. Passing a hypercube through the 3D world would also create cross sections, such as triangular prisms, hexagonal prisms and dodecahedrons. This is a similar (though slightly different) idea to the shadow of the hypercube I presented in the last blog post. This concept is applied in a more fantastical (and terrifying) way in the passage.
A Bit More About Matt Parker (and other resources!)
In his book, Things to Make and Do in the Fourth Dimension, Matt Parker provides many more examples of the fourth dimension and even some projects to better explain it, such as showing you how to build your own 4D cube. He also diagrams many of these concepts. He discusses many other topics in addition to the fourth dimension, and I highly recommend his book. There's also a science fiction story (Matt Parker's inspiration for this section) where a 4D protagonist attacks a 3D victim. It was written by a famed comic-book artist, Alan Moore, it is from "1963 - Tales of the Uncanny" and it is called "It Came from ... Higher Space!" if you'd like to check it out.
If you'd like to see more of Matt Parker, he has several math comedy routines on YouTube: https://youtu.be/UBX2QQHlQ_I, https://youtu.be/Bcr9-93wXng, https://youtu.be/RP8jepN3zMc. His YouTube channels are standupmaths and Matt Parker. He also frequently makes an appearance on Numberphile, another amazing YouTube channel for recreational math.
Here is his most (in)famous video: https://youtu.be/aOT_bG-vWyg ☺
And here is a more serious video that dives into a nice concept, a card trick: https://youtu.be/l7lP9y7Bb5g.
He has many more videos where those came from and I'd advise you to check them out.
I can't stress enough just how much I love Numberphile and Matt Parker's channels. They show the side of math that should be revealed more often: the fun, light-hearted, enjoyable side, and Matt Parker's routines inevitably make me cry with laughter. Numberphile has almost 3 million subscribers, which is huge for a channel in its domain. These channels were a large part of how I was able to develop my passion for math, and I owe a lot to them. Seriously, I can't endorse them enough. Go check them out.
Back to the Fourth Dimension! The second resource I'd like to show you is a video with Neil deGrasse Tyson who explains how Monster's Inc is related to the fourth dimension. It's a fascinating discussion about wormholes, storage and the like. It's one of the best explanations of the fourth dimension I've seen:
The third resource I'd like to show you is by YouTuber Eva Smerekancyh. She is currently a freshman at MIT, where she plans to study astrophysics and is also an incredibly accomplished artist. This video is claymation and does a wonderful job of diving into the fourth dimension.
That concludes my discussion of the fourth dimension. I hope you learned something enjoyable in the process! Give me some feedback below in the comments if you'd like. Do you have any questions about the fourth dimension? Or about anything math-related? Leave those as well.
I'll summarize by saying: The fourth dimension is a fascinatingly elusive and abstract concept that is difficult for us to wrap our brains around. With some resources and explanations, we can begin to build the intuition and understand the beauty of a mathematical concept that exists primarily outside of our intuition. Despite this, the fourth dimension is remarkably practical as it is intertwined in the big data world we live in, though I mentioned it primarily here to reveal something intellectually profound for fellow puzzle lovers.
If you'd like to learn more about the fourth dimension, just let me know in the comments! I've actually learned a lot more about higher dimensions since I first drafted these blog posts, and I can write a third installment if you would enjoy it!