The Fourth Dimension

On the off-chance anyone else comes along for the Carnival… Sometimes, when I’m asked what mathematicians do, I’ll start talking about something related to geometry in more than three dimensions. One question I occasionally get is “But what is the fourth dimension?” (The more physics literate will say something like “ I know that time is the fourth dimension, but what would be the fifth dimension?”) I usually try to explain that higher dimensions are abstract concepts, and that we understand them through analogies.

Along those lines, here are two facts about the fourth dimension that seemed inexplicable to me when I first heard them, but now seem obvious:

  1. In three dimensions, if two planes intersect they must intersect in a line. In four dimensions, two planes can intersect at a single point.
  2. In three dimensions, if you try to roll up a piece of paper into a torus, you have to crinkle the paper to close up the tube into a torus. In four dimensions, you could do it without crinkling the paper.