Our imagination can take as to the furthest lands, and what can be further than other or higher dimensions? The problem with traveling to higher dimensions with an imaginary multi-dimensional rocket is that even thinking about higher dimensions is hard (we will exclude time as the fourth dimension for a while). This is nicely illustrated in a well-known scientific joke:
A mathematician and his best friend, an engineer, attend a public lecture on geometry in thirteen-dimensional space. “How did you like it?” the mathematician wants to know after the talk. “My head’s spinning”, the engineer confesses. “How can you develop any intuition for thirteen-dimensional space?” “Well, it’s not even difficult. All I do is visualize the situation in arbitrary N-dimensional space and then set N = 13.”
To make it possible, better to take it one step at a time. Check for introduction, and look for ways to simplify the journey. We want to strengthen our imagination, and for that, we will search for aids that will help us to imagine and reason about higher dimensions. We recommend starting with something familiar – like the tesseract.
Many will probably think that we mean The Tesseract, a powerful artifact that appears in the Marvel Cinematic Universe and has an unlimited source of power of alien origins, related to the Infinity Stones. It’s awesome, but barely related to the geometrical tesseract that we will discuss here (although, after reading this article you may find some references to the geometrical tesseract in these movies – look for them).
In geometry, the tesseract, also known as the hypercube, is a four-dimensional cube. Since cubes are familiar to all, they can be a good aid to reason about four-dimensional worlds. On the other hand, to some it may come as a surprise that both of the following two images are cubes:
No, we haven’t lost our mind or tried any of Alice’s favorite mushrooms. As some may have figured, the dot is a “cube” in a zero-dimensional world. It will make sense in a second.
First, pay attention that what we see in the right image isn’t really a cube, but a representation of a cube on a flat, two-dimensional screen. That’s why the red lines intersect with the green lines – there is no “depth” in the image, so we do the best we can to schematically draw a cube under these circumstances. It’s called the projections of the three-dimensional cube on two-dimensions. The green-blue squares don’t look like squares, but like parallelograms – it’s the limitation of the two-dimensional medium, but we all understand that these parallelograms are here to represent squares.
Next, to understand what a four-dimensional cube is, it’s better to think first about simpler cases. Like, zero-, one-, two- and three-dimensional worlds.
There are lots of resembles between the cube and a square – they are relatives. The cube is made of a couple of parallel squares, connected in a way that creates another pair of parallel squares.
The square is made of a couple of parallel lines, connected in a way that creates another pair of parallel lines. Stretching this resembles a bit further, we can say that a dot is a zero-dimensional cube; by copying it, moving aside, and connecting to the original dot, we create a line segment, which is a one-dimensional cube; by repeating that, copying the line segment, moving it aside (to the same distance as the segment’s length) and connecting it, we get a square; If we copy the square and connect, we get a cube, and if we copy a cube and connect, we get… right, a tesseract, the hypercube.
The cube is built from squares, but the tesseract is built from cubes. It’s hard to see the cubes in the image, but that’s the limitation of a two-dimensional projection. By rotating the tesseract, we can look at it from different perspectives:
Now that you know how to build a four-dimensional cube, you can also imagine (and event draw) five-, six-, seven- and higher-dimensional hypercubes. Just follow the same method: Copy the lower-dimensional cube, connect, and voilà!
It may be surprising, but hypercubes are actually used in real-life. High-dimensional cubes are used to design the connection of multiple computers in a network, or in Gray codes, invented by Frank Gray, that are widely used to prevent spurious emission from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems.
Now you have another tool for your multi-dimensional adventure, and we advise you to practice it and stretch further your imagination. Before you go, we have just one question for you: Can you do the same with a triangle? Hint: Look for a “simplex”.
Featured image by Jason Hise, Public Domain.
Related Articles: