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So, if we are in $\Bbb R^{n}$ there are $n(n-1)/2$ rotations. Which is to say:

Given $1$ of the $n$ axis, we can rotate it onto $n-1$ other axis. We then divide by $2$ to not overcount.

That's OK, however if we are in a 2D plane, rotating $x$ axis onto $y$ axis means rotating the plane around a $z$ axis which is outside the 2D space we are considering and inside $\Bbb R^{3}$.

If we instead rotate one of the 3 axis in $\Bbb R^{3}$, all the rotations are around axis inside the space we are considering.

Why this? Does this happen even in higher dimension? Is it a quirk of 3D spaces and a few others, or is it instead a quirk of 2D space?

Daniele Tampieri
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Matteo
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  • In $\mathbb R^2$ there is one rotation. Proof: $2(2-1)/2=1,.$ – Kurt G. Aug 28 '23 at 09:12
  • The way you've asked this question is a bit confusing. You start talking about the number of ways to choose two axes from $n$, which is obviously $\binom{n}{2}$. But it seems you're actually asking about which dimension axes of rotations lie in, is that right? – Chris Lewis Aug 28 '23 at 09:24
  • Yes, there is one rotation Kurt. But as I said the axis around which we are rotating is out of the 2D plane. However in 3D the axis around which we rotate are not out of the 3D plane. That's what I'm asking Chris – Matteo Aug 28 '23 at 10:39
  • A rotation in 2D is not characterized by an axis. In 3D it is one axis. In 4D it is two axes. – Kurt G. Aug 28 '23 at 11:16
  • I don't understand. If I rotate clockwise X and Y axis in 2D, i rotate around an axis perpendicular to that plane. Am I wrong? – Matteo Aug 28 '23 at 12:04
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    That's not wrong. If you want to visualize it that way embed your $xy$-plane into $\mathbb R^3,.$ BTW it is still very unclear what that question is about. I claim that in $\mathbb R^n$ you need $n-2$ axes to characterize your rotation. – Kurt G. Aug 28 '23 at 12:14
  • Related? https://math.stackexchange.com/questions/2292611/visualizing-rotation-in-even-dimensions – Andrew D. Hwang Aug 28 '23 at 14:19

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if we are in a 2D plane, rotating X axis onto Y axis means rotating the plane around a Z axis which is outside the 2D space we are considering and inside $R^{3}$.

No it doesn't. That's one way to think about the rotation, especially since we are so used to working in $3$ dimensions physically ourselves (and thinking about e.g. wheels and axles), but you don't actually have to talk about this third axis if you don't want to. A rotation in $\mathbb{R}^2$ is simply a rotation around the origin, and that's it.

Qiaochu Yuan
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  • Using your logic we can make 3 different rotations around the origin in $R^{3}$ and that's right. However how can we distinguish one from the another? I simply use the axis around which they implement the rotation. In $R^{3}$ it happens all these axis belong to $R^{3}$. In$R^{2}$ it happens the axis around which we implement the rotation doesn't belong to $R^{2}$. That's my (dumb) question. I may answer myself: the rotation regards a 2D plane, so you need an additional axis if you are rotating all the 2D space and you don't need any additional axis if you rotate a subspace of 3D. – Matteo Aug 29 '23 at 10:36
  • @Matteo: you can just specify the plane that the rotation takes place in. In $\mathbb{R}^3$ it happens that you can specify a plane by specifying a vector normal to it (that's the axis of rotation) but that isn't true in higher dimensions and it isn't necessary in $\mathbb{R}^2$. In higher dimensions a general rotation takes place along several planes simultaneously, e.g. in $\mathbb{R}^4$ a general rotation involves two planes. – Qiaochu Yuan Aug 29 '23 at 18:19