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

2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Example $2.1.7$, Ponderable $2.1.7$