# 28-30.01.14 Geometric algebra

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On Tuesday January 28 Mr. Gunn continued his discussion of the geometric algebra of $\mathbb{R}^2$.  He showed that reflections and rotations can be written as sandwich operators.  For example, reflection of the vector $\mathbf{u}$ in the vector $\mathbf{v}$ can be written $\mathbf{v}\mathbf{u}\mathbf{v}$. This is a sandwich, whose bread is $\mathbf{v}$ and whose meat is $\mathbf{u}$. By concatenating two reflections one obtains another sandwich operator for a rotation around the origin by an angle $\alpha$.

To obtain this rotation formula, define $\mathbf{R} := \mathbf{v}_2 \mathbf{v}_1$ be the geometric product of two unit vectors which meet at an angle of $\dfrac{\alpha}{2}$, CCW from $\mathbf{v}_1$ to $\mathbf{v}_2$. Then we showed in the previous lecture that $\mathbf{R} = \cos{\dfrac{\alpha}{2}} + \sin{\dfrac{\alpha}{2}}\mathbf{I}$.  (Recall that $\mathbf{I} = \mathbf{e}_1 \mathbf{e}_2$ is the pseudoscalar of the algebra).   Recall  that the concatenation of two reflections is a rotation around the meeting point of the lines, through twice the angle between the lines. Then the desired rotation can be written as the concatenation of reflections in these two vectors: $\mathbf{v}_2(\mathbf{v}_1 \mathbf{u} \mathbf{v}_1)\mathbf{v}_2$.  Since the geometric product is associative we can reparenthesize to obtain $(\mathbf{v}_2 \mathbf{v}_1 ) \mathbf{u} (\mathbf{v}_1\mathbf{v}_2)$.  Finally,  let $\mathbf{\widetilde{R}} = \mathbf{v}_1 \mathbf{v}_2$ denote the reversal of $\mathbf{R}$, the element obtained by reversing the order of all geometric products in $\mathbf{R}$.   Then the formula for the rotation reads \mathbf{R}\mathbf{u}\mathbf{\widetilde{R}}$. The element$\mathbf{R}\$ is called a rotor.