# Defending the fourth dimension

The sad events around Boston these last two days have been so overwhelming that the event I was going there for has been postponed. So I find myself with an extra day in New York.

It’s important to use that sort of extra time productively. For example, thinking about interesting ideas.

Ever since my blog post about the pyramid of spheres, I have found myself thinking about how beautifully those shapes fit together. And I remembered that when I was a kid, I was completely fascinated to discover that exactly six quarters can be fit around a quarter — to make a perfect hexagon. To me this was pure magic.

You can sort of do something like this in three dimensions, since twelve spheres can fit perfectly around a sphere (I once made an on-line visualization of this). But it’s not as pretty in three dimensions — the resulting arrangement doesn’t form a perfectly regular structure.

Yet it turns out that in four dimensions it all becomes perfect again. Just as in two dimensions you can fit six circles perfectly around a circle, which extends into a perfectly regular pattern that goes on forever, in four dimensions you can fit twenty four “hyperspheres” (spheres in four dimensions) perfectly around one hypersphere, which also extends into a perfectly regular pattern that goes on forever. The beautiful four dimensional shape that does this is called a “24-cell”.

I know 4D is a touchy subject for many people. After all, we can’t actually visit the fourth dimension. But hey, the magnificent thing about being human is that we can think about things that don’t physically exist. These things include the very words that come out of our mouths, as well as love, honor, joy, happiness, and a sense of humor.

In fact, just about all of the things we care about most deeply are things that don’t physically exist. So what’s wrong with thinking about the fourth dimension?