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I'm trying to rotate a 4 dimensional point (w,x,y,z). So far I've been rotating around planes (wx,xy,yz,zw,wy, and xy), but the order in which I do these rotations changes the results and can sometimes result in gimbal lock.

I understand that quaternions are the solution to this problem in 3 dimensions, but can quaternions also do rotations in 4 dimensions and if so, how?

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    @Jyrki I see -- thanks for the link. I think I understand the problem better now.2011-06-12

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As Jyrki suggests, it is possible to use a pair of unit quaternions to describe a rotation is four dimensions. Specifically, any rotation $R$ can be written as $ R(v) \;=\; avb $ where $v$ is a vector in $\mathbb{R}^4$ (treated as a quaternion), and $a$ and $b$ are the unit quaternions describing the rotation.

Given two such rotations R(v) \;=\; avb \qquad\text{and}\qquad R'(v)\;=\;a'vb' the composition R\circ R' (i.e. rotating R' and then rotating $R$) is obtained by multiplying the corresponding quaternions: (R\circ R')(v) \;=\; (aa')v(b'b).

Rotations around the six coordinates planes can be described as follows: $ \begin{array}{cc} R_{wx}^\theta(v) \;=\; e^{-i\theta/2}ve^{i\theta/2} & R_{yz}^\theta \;=\; e^{i\theta/2}ve^{i\theta/2} \\ \\ R_{wy}^\theta(v) \;=\; e^{-j\theta/2}ve^{j\theta/2} & R_{xz}^\theta \;=\; e^{j\theta/2}ve^{j\theta/2} \\ \\ R_{wz}^\theta(v) \;=\; e^{-k\theta/2}ve^{k\theta/2} & R_{xy}^\theta \;=\; e^{k\theta/2}ve^{k\theta/2} \end{array} $ where $e^{j\theta} \;=\; \cos(\theta) + j\sin(\theta)$, etc.

Finally, note that the quaternion-pair representation of a rotation is not unique. Specifically, the rotation with coefficients $(a,b)$ is the same as the rotation with coefficients $(-a,-b)$, for all unit quaternions $a$ and $b$.

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    Does that mean, I can take any two unit quaternion $a,b$ and make a rotation $R=avb$ which will generate a rotation part of $SO(4)$. So the set of all unit quaternion pairs span the set $SO(4)$?2016-03-07
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No, I don't believe quaternions work in 4 dimensions. See this and this.

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    Doesn't the double cover map $S^3_L\times S^3_R\rightarrow SO(4)$ described in your latter link allow Greg to avoid gimbal locks much the same way that the use of the covering map from the unit quaternions to $SO(3)$ does? I may be completely wrong about this. I'm just curious :-)2011-06-12