Rotation of Unit Cube about Vertical Axis

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Example of Symmetry in context of Geometry

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

Let the vertices of $C$ be defined as:

\(\ds O\) \(=\) \(\, \ds \tuple {0, 0, 0} \, \) \(\, \ds =: \, \) \(\ds dbl\)
\(\ds P_1\) \(=\) \(\, \ds \tuple {1, 0, 0} \, \) \(\, \ds =: \, \) \(\ds dfl\)
\(\ds P_2\) \(=\) \(\, \ds \tuple {0, 1, 0} \, \) \(\, \ds =: \, \) \(\ds dbr\)
\(\ds P_3\) \(=\) \(\, \ds \tuple {0, 0, 1} \, \) \(\, \ds =: \, \) \(\ds ubl\)
\(\ds P_4\) \(=\) \(\, \ds \tuple {1, 1, 0} \, \) \(\, \ds =: \, \) \(\ds dfr\)
\(\ds P_5\) \(=\) \(\, \ds \tuple {1, 0, 1} \, \) \(\, \ds =: \, \) \(\ds ufl\)
\(\ds P_6\) \(=\) \(\, \ds \tuple {1, 1, 0} \, \) \(\, \ds =: \, \) \(\ds ubr\)
\(\ds P_7\) \(=\) \(\, \ds \tuple {1, 1, 1} \, \) \(\, \ds =: \, \) \(\ds ufr\)


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:

\(\ds C_0\) \(=:\) \(\ds \set {O, P_1, P_2, P_3, P_4, P_5, P_6, P_7}\) the set of $8$ vertices of $C$
\(\ds C_1\) \(=:\) \(\ds \set {uf, ur, ub, ul, fr, br, bl, fl, df, dr, db, dl}\) the set of $12$ edges of $C$
\(\ds C_3\) \(=:\) \(\ds \set {F, B, U, D, L, R}\) the set of $6$ faces of $C$


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:

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


Sources

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