From Flatland to the fourth dimension (part 6)

Sorry, “law and order” series—this set of posts about the fourth dimension is now the longest, at 6 posts. Today I will cover the shape of the universe, wheels, and paper in four dimensions. Again, if you have not read the previous posts in this series, with the exception of #1, today’s post is going to seem even more confusing than it already is.

First, I would like to go into the shape of the universe in relation to the fourth dimension. Though there are several attempts to explain the shape of the universe, none of them have been completely proven, and the universe’s true shape is still unknown. It is possible that even if the universe is three dimensional is it “curved” four dimensionally, such as into the surface of a 4D sphere of cube. This seems difficult to visualize, but try an analogy: imagine a 2D ant living on the surface of a piece of paper. The paper can be curved, bent, or even folded but that ant would not know about it. Likewise, our universe can be curved in the same way. However, this might have some effects on exploration of the universe or our attempts to survey it. For example, our universe could be shaped like the surface of a 4D torus (doughnut). This is very hard to visualize, but think back to the analogy for a moment: consider what would happen if the 2D ant’s world were curved into the surface of a doughnut. The ant could walk in a straight line and actually come back to where he started (in several different ways—look at the shape of a torus).

Next, paper. In 3D paper is actually 3D as well, but for practical purposes it is 2D because we do not right on the “sides” of a piece of paper—not enough room. In 4D, then, paper is effectively three-dimensional. On a square 2D piece of paper, there are three paths from one corner to the others; on a 3D one there are six. Also, there are only four ways to cut a piece of paper symmetrically in half in three dimensions—in four dimensions, there are 8.

Lastly, wheels. A wheel in four dimensions is similar to a 4D torus, similar to the way a 3D tire resembles a torus. The area of a 4D tire that touches the ground, unlike a 3D one, is a plane; this means that if a 4D tire were rolling across the surface of our 3D world it would appears as if a thin rectangular prism were moving across the room. A 4D wheel as four directions to fall over, unlike a 3D tire, which only has two. Also, a four dimensional spherical tire (the 4D equivalent of a cylinder) has similar properties to a 3D cylinder—it has two directions to fall over into, but four to roll into. I realize all of this is almost impossible to visualize, but give it a try anyway.

This concludes the series on the fourth dimension for the time being. Tomorrow I would like to talk about some pressing current events that have happened over the past week.