Last week we introduced the tesseract, which is to a cube what a cube is to a square — an extension into one more dimension.  That’s why it’s also called a hypercube.  The first tesseract diagrams I ever saw were so confusing — they looked like lots of overlapped squares tied together with lines that didn’t make much sense.  I wondered, “Wouldn’t it be easier to understand a tesseract if I could see it rotating?”

Years later computers and I had both moved ahead to where I could generate the pictures you see in this post.  What I learned while doing that was that 4-D figures have two equators.  In four dimensions, it’s possible for something to rotate in two perpendicular directions at the same time.  Read on and please don’t mind my doggerel — it doesn’t bite.

	The LINE is just a single stroke, a path from here to there.
Stretch it out beside itself and you will have a SQUARE. Where’s its face when it turns around? Gone, ’cause its back’s not there.
	The CUBE’s a square made thick, you see. Length, breadth and depth comprise a full 3-D.
Add yet a thickness more, crosswise all to X, Y, Z. A TESSERACT on a corner spins but an XY-slice is all we see.
	But the axis, too, can rotate through a path that’s drawn invisibly. Four faces grow and shrink in place — it’s hard to do that physically.
This tesseract is tumbling ’bout two equators perpendicular. Were I in such a state, I vow, I’d be giddy, even sickular.

In the 4-D views, when one of the tesseract’s cubical faces appears to disappear into an adjacent face, what’s actually happening is that the face is sliding past the other face along that fourth dimension (which I called W because why not?)

You’re looking at a two-dimensional picture of the three-dimensional projection of a four-dimensional object as it moves in 5-space (X, Y, Z, W, and time — if it didn’t move in time then it couldn’t be spinning).

Next week — Herr Klein’s bottle, or rather flask, or rather surface.

~~ Rich Olcott