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- Apr 14, 2013

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Let $\displaystyle{W:=\left \{\begin{pmatrix}x\\ y\\ z\end{pmatrix}\in \mathbb{R}^3\mid x,y,z\in \{-1,1\}\right \}}$.

Draw the set $W$ in a coordinate system. Let $v=\neq w$ and $v,w\in W$. If they differ only at one coordinate connect these points by a line.

With this description we get a cube with vertices $(\pm 1 , \pm 1 , \pm 1)$, right?

Let \begin{equation*}d:=\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\ , \ s:=\begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \ , \ D:=\left \{\begin{pmatrix}e_1 & 0 & 0 \\ 0 & e_2 & 0 \\ 0 & 0 & e_3 \end{pmatrix} \mid e_i\in \{-1,1\}\right \}\end{equation*}

Show that for all $a\in \{d,s\}\cup D$ the map $w\mapsto aw$ is a symmetry of the cube and descibe the symmetry gemetrically.

Do we have tomultiply all vertices of the cube with each of these matrices? Or what are we supposed to do?

The following moves are the symmetries of the cube $W$:

1. Rotation at the axis $\mathbb{R}\begin{pmatrix}0 \\ 0\\ 1\end{pmatrix}$ with angle $\frac{\pi}{2}$.

2. Rotation at the axis $\mathbb{R}\begin{pmatrix}0 \\ 1\\ 0\end{pmatrix}$ with angle $\frac{\pi}{2}$.

3. Rotation at the axis $\mathbb{R}\begin{pmatrix}1 \\ 1\\ -1\end{pmatrix}$ with angle $\frac{2\pi}{3}$.

4. Rotation at the axis $\mathbb{R}\begin{pmatrix}1 \\ -1\\ -1\end{pmatrix}$ with angle $\frac{2\pi}{3}$.

5. Reflections to the plane, that is spanned by the vectors $\begin{pmatrix}1\\ -1\\ 0\end{pmatrix}$ and $\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}$.

How can we find for these symmetries a matrix $a$ so that the map $w\mapsto aw$ describes each symmetry?