# Rotations of 4-Cubes

I have recently learned the orbit stabilizer theorem, and have encountered unexpected results pertaining to the rotations of a tesseract; I am curious if there is any intuition for this.

A $4$-Cube has $(16*4)/2 = 32$ edges using some simple counting, and has $24*8 = 192$ rotational symmetries, which I obtained by considering that each of the $8$ cubical 'faces' can map to any other in $24$ ways (each way is a rotational symmetry of a 3-cube). Each edge has all $32$ edges in its orbit, and so by orbit stabilizer we conclude each edge has $192/32 = 6$ stabilizing rotations.

This seems very bizarre to me, as it suggest that you can perform $3$ rotations of a $4$-cube that keep any edge in the same position, facing the same direction, so essentially nothing has changed. Does this mean it is possible to rotate a $4$-cube non-trivially in $2$ ways while keeping one edge completely fixed? Or does the edge have to move, only to end up exactly as it was when it began?

Is my reasoning correct, and is there any intuitive way to understand these results? In general, are rotation in higher dimensions too bizarre to have an intuitive grasp of, and are there some interesting things I should know about these rotations? Thanks!

• I think you've stumbled upon one good reason people tend to think in terms of Coxeter/reflection groups :) – pjs36 Nov 23 '15 at 1:14
• I was going to attempt to answer, but instead I'll just send two links. This post by John Baez has some very nice remarks (that don't answer your question) and a link to the Wikipedia article with all the gory details on rotations in $4$-space. If I manage to make anything comprehensible, I'll post an answer, but that's at least something, for now (and possibly ever, from me!). – pjs36 Nov 23 '15 at 4:28
• Interesting, thanks! But I still think I'm quite a ways from a firm understanding of 4+ dimensional rotations :) – B Gunsolus Nov 23 '15 at 16:25

## 1 Answer

If you examine a tesseract, you should be able to find three cubical hyperfaces that meet at a single edge. In fact, every edge has three hyperfaces arranged symmetrically around it in this fashion.

While keeping this edge fixed, then, you can rotate one of the three adjoining cubical hyperfaces onto one of the others. Repeat this rotation to find the second non-trivial rotation that leaves this edge fixed. The third repetition takes the hyperface back to its original position.

• This makes sense, thanks! I suppose a 2D creature would struggle to understand how a 3-cube has rotational symmetries that keep a vertex unmoved, and my problem was analogous. – B Gunsolus Nov 23 '15 at 16:24
• That analogy occurs to me, too. You might enjoy Flatland, by E. A. Abbott. The main character of that book is a 2D creature struggling to understand a third dimension. – David K Nov 23 '15 at 20:48
• I have read flatland :) Interesting stuff. – B Gunsolus Nov 23 '15 at 22:26