## 4-dimensional hypercube

I’ve been reading Michio Kaku’s Hyperspace, and it’s got me trying to visualize the fourth spacial dimension. It’s not possible to do, but it’s fun to try. Fortunately, the internet has plenty of videos on the matter, a few of which I’ll present here.

As the video I embedded in a previous post about visualizing higher dimensions said, sometimes it’s easier to imagine a higher dimension beyond the three we’re familiar with by thinking of the higher dimension as a dimension you “fold lower dimensions through” to get a desired result. For instance, as shown in the video, folding a 2-dimensional sheet through the third dimension allows the edges of the sheet to touch, so an ant can crawl from one edge to the other. If you lived on the 2-D sheet and could only see in two dimensions, it would appear to you that the ant disappeared from one edge and instantly reappeared on the other. We can’t visualize dimensions higher than three, but we can visualize how actions in these higher dimensions would would look in our 3-dimensional world, analogous to a creature who can only see in two dimensions watching an ant disappear from one place and reappear in another.

A popular 4-dimensional object to try to visualize is a tesseract, which is a 4-dimensional hypercube. We can’t picture it, but we can picture it’s projection, or shadow, in three dimensions. Here is a video of the projection of a 4-D cube rotating:

Here’s another video showing the construction of a hypercube:

It shows three dimensions of the tesseract being built on the left side, and one of those three dimensions along with a fourth dimension on the right side. We can’t picture the whole hypercube, but we can picture separate parts of the cube and piece together how they interact from those visualizations. I had to watch it a couple times to really piece together what’s going on, but I think it’s worthwhile.

This page has some really neat videos showing cubes folded up. The first one shows 2-D squares (in red) folded up to create a 3-D cube:

The pink squares are the cube’s 2-D shadow as it’s being folded and unfolded.

We can’t picture a 4-D cube, but an unfolded tesseract in 3-dimensions would look like this:

The second video shows what a 4-D cube’s 3-D shadow looks like as it’s being folded:

And now that you mind has been suitably bent, I’ll leave with with an image of Salvador Dali’s Christus Hypercubus: