Is there a higher dimensional analogue to translation and rotation. Translation occurs along an axis, and rotations occurs along a plane. Is there some isometry that occurs along a 3-plane (hyperplane?) and does it have a name? Do these isometries generalize well? That is, are there $\binom{N}{n}$ different $n$-dimension isometries in $N$ space where a rotation would be a 2-dimension isometry?
Higher Dimensional analogue to translation and rotation.
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linear-algebra
geometry
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0Check out [Quaternions](http://en.wikipedia.org/wiki/Quaternion). Quaternion multiplication is rotation in $4$ dimensions similar to how complex multiplication is rotation in $2$ dimensions. This can be generalized to $n$ dimensions using matrices – 2012-10-23
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Rotations can occur in $n$-dimensions in the same way. For example, you can have rotation on a sphere, for the case $n = 3$.
In general, these $n$-dimensional rotations are characterized by elements of the special orthogonal group, denoted $SO(n)$, which consist of the set of $n \times n$ matrices $Q$ such that $Q^T Q = I$ and $\det(Q) = 1$. For any point $x \in \mathbb{R}^n$, acting such a rotation on $x$ is given by considering the matrix-vector product $Qx$.
If you want such a rotation $Q$ to have as part of its behavior rotation when restricted to some $2$-dimensional plane, then you just want a $Q = Q_1 \oplus Q_2$, where $Q_1$ is a $2 \times 2$ rotation matrix.