Rotation in 4D, number of axes required

Question

I've noticed that rotation can be interpreted in 2 different manors. Does rotation occur about an axis, or through a pair of axes?

What I would like to know is in 4D are there 4 fundamental rotations (one about each axis) or are there 6 fundamental rotations (one about each pair of axes)?

If one of these interpretations is rotation, what is the name for the other?

score 2 · Accepted Answer · answered Feb 27 '20 at 13:38

2

A rotation always acts on a plane. So there are 6 different rotations in $\mathbb{R}^4$ that keep two axis fixed and rotate the plane spanned by the other two. You can't rotate around an axis unless you are in $\mathbb{R}^3$. To see this consider the plane $\mathbb{R}^2$, there is only one rotation of the entire plane, not two for the two axis.

answered Feb 27 '20 at 13:38

quarague

5,921

Of course... The pattern is ${1,3,?}$. Not ${2,3,?}$ Thank you! – Ben Crossley Feb 27 '20 at 13:42
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btw the number of independent rotations is $d(d-1)/2$, where $d$ is the dimensionality. It's the number of degrees of freedom of antisymmetric matrix $d\times d$ – Vasily Mitch Feb 27 '20 at 13:51

Rotation in 4D, number of axes required

1 Answers1