Rotation of Unit Cube about Vertical Axis

Example of Symmetry Mapping
Let $C$ be the unit cube embedded in real Cartesian space of $3$ dimensions as follows.

Let the vertices of $C$ be defined as:

Thus:
 * $u$ and $d$ stand for up and down respectively
 * $f$ and $b$ stand for front and back respectively
 * $l$ and $r$ stand for left and right respectively.

Let:

Let $r$ be the rotation by $180 \degrees$ about the axis defined by the line through the points $\tuple {\dfrac 1 2, \dfrac 1 2, 0}$ and $\tuple {\dfrac 1 2, \dfrac 1 2, 1}$.

This is a symmetry of $C$ which induces $3$ permutations:


 * $f_0: C_0 \to C_0$


 * $f_1: C_1 \to C_1$


 * $f_2: C_2 \to C_2$

$f_0$, $f_1$ and $f_2$ can be specified explicitly as follows:


 * $\begin{array} {|r|r|}

\hline v & \map {f_0} v \\ \hline O = dbl & P_4 = dfr \\ P_1 = dfl & P_2 = dbr \\ P_2 = dbr & P_1 = dfl \\ P_3 = ubl & P_7 = ufr \\ P_4 = dfr & O = dbl \\ P_5 = ufl & P_6 = ubr \\ P_6 = ubr & P_5 = dfl \\ P_7 = ufr & P_3 = ubl \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline e & \map {f_1} e \\ \hline uf & ub \\ ur & ul \\ ub & uf \\ ul & ur \\ fr & bl \\ br & fl \\ bl & fr \\ fl & br \\ df & db \\ dr & dl \\ db & df \\ dl & dr \\ \hline \end{array} \qquad \begin{array} {|r|r|} \hline f & \map {f_2} f \\ \hline F & B \\ B & F \\ U & U \\ D & D \\ L & R \\ R & L \\ \hline \end{array}$