From Flatland to the fourth dimension (part 3)

Last time, I explained the concept of the fourth spatial dimension and a few applications of it. Today, I would like to discuss more of what the fourth dimension would be like. I advise against reading today’s post without reading yesterdays, and reading part 1 of this series is also advisable, though not as necessary.

Yesterday I explained how we can deduce certain aspects of the fourth dimension by analogy. This is precisely what I would like to do today: explore what it would be like to experience four dimensions by looking at what it be like to experience three after living in two.

I would first like to talk about four-dimensional beings. A four-dimensional being would obviously be in the shape of a 4D figure, and it would be able to move in all 8 perpendicular directions, at least in zero gravity (like the way we can move in all 6 directions in zero-g). This obviously hints at the structure of a 4D being—it must have at least 4 legs to be stable (humans are not stable—we would need 3 legs in a tripod). But, like human beings, it is possible to get away with less—but I doubt a 4D being could have less than 3 because there are now more directions to “fall” (fall over, that is). Also, a 4D being’s line of sight would be a cube. This is because a 2D being’s is a line (recall the explanation of how “Flatlanders” see in part 1 of this series) and human beings see 2-dimensionally. However, because we have depth perception we can distinguish 3D objects. But our field of view is still a plane—our vision is akin to a photograph; everything is reduced to a 2D image. Thus, a 4D being would see in terms of a cube; a 4D photograph would be three-dimensional. At first this sounds normal enough, but consider the implications: 2D vision grants us the ability to see all the sides and the inside of an opaque square at the same time—a 4D being would be able to see all the sides, and the inside, of an opaque cube at the same time. Therefore, from the correct vantage point, a four-dimensional being could look at a human and see every inch of his skin as well as the inside and outside of all of his organs.

Next, consider what happens when four-dimensional beings or objects interact with our three-dimensional plane. This, too, can be done by analogy. In Flatland, the three-dimensional character reaches in and touches the insides of a two-dimensional character; this is the equivalent of touching the inside of a square drawn on a piece of paper. A 4D beings would have similar powers, and would be able to reach inside of sealed 3D objects or beings.

Also, consider what happens when a 4D object is passed though our 3D plane. This, too, can be explained by using an analogy from Flatland. When a square passes through a two-dimensional plane, a 2D observer sees a line appear and then later disappear. The observer can only perceive only one “square” at time (think back to when I described a cube as a stack of paper squares). Likewise, when a sphere is passed through a 2D plane an observer would see a line appear, grow in size, and then shrink and disappear. (Again, think back to part 2.) Similarly, if a hypercube is passed through our 3D plane we would see a cube appear and then disappear, since we can only see one cube at a time. If a glome (4D sphere) was passed through our 3D plane, we would see a sphere appear and grow in size and then shrink back down to nothing again. This is a bit harder to visualize, but it is essentially the same: the glome is made up of an infinite number of spheres, but we can see only one at a time because of the way they are stacked. 

I would also like to talk about geometrical nets. A net is a shape of n-1 dimensions that can be folded into an object of n dimensions. The net of a cube, for example, is 6 squares in the shape of a cross; these can be folded into a box. Interestingly, the net of a hypercube is 8 cubes arranged into a 3D cross: 4 cubes are stacked vertically, and the other 4 are attached to the other 4 exposed sides of the cube second from the top. The nets of other 4D figures are also 3D figures. However, though we can make the nets, we cannot fold them, since they must utilize the four-dimensional 8 perpendicular directions in order to fold. Note that once folded, one cube of the net of a hypercube remains in our 3D plane; the others will appear to have simply vanished because they are outside of our plane. In an amusing short story by Robert A. Heinlein, an architect builds a building that is the net of a hypercube, and an earthquake causes the building to fold into the 4D shape. This is impossible, of course, but the concept is clever.

Tomorrow, I will cover even more aspects of the fourth dimension as extrapolated by analogy.