# Rotation of Unit Cube about Vertical Axis

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## 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:

\(\displaystyle O\) | \(=\) | \(\, \displaystyle \tuple {0, 0, 0} \, \) | \(\, \displaystyle =:\, \) | \(\displaystyle dbl\) | |||||||||

\(\displaystyle P_1\) | \(=\) | \(\, \displaystyle \tuple {1, 0, 0} \, \) | \(\, \displaystyle =:\, \) | \(\displaystyle dfl\) | |||||||||

\(\displaystyle P_2\) | \(=\) | \(\, \displaystyle \tuple {0, 1, 0} \, \) | \(\, \displaystyle =:\, \) | \(\displaystyle dbr\) | |||||||||

\(\displaystyle P_3\) | \(=\) | \(\, \displaystyle \tuple {0, 0, 1} \, \) | \(\, \displaystyle =:\, \) | \(\displaystyle ubl\) | |||||||||

\(\displaystyle P_4\) | \(=\) | \(\, \displaystyle \tuple {1, 1, 0} \, \) | \(\, \displaystyle =:\, \) | \(\displaystyle dfr\) | |||||||||

\(\displaystyle P_5\) | \(=\) | \(\, \displaystyle \tuple {1, 0, 1} \, \) | \(\, \displaystyle =:\, \) | \(\displaystyle ufl\) | |||||||||

\(\displaystyle P_6\) | \(=\) | \(\, \displaystyle \tuple {1, 1, 0} \, \) | \(\, \displaystyle =:\, \) | \(\displaystyle ubr\) | |||||||||

\(\displaystyle P_7\) | \(=\) | \(\, \displaystyle \tuple {1, 1, 1} \, \) | \(\, \displaystyle =:\, \) | \(\displaystyle ufr\) |

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:

\(\displaystyle C_0\) | \(=:\) | \(\displaystyle \set {O, P_1, P_2, P_3, P_4, P_5, P_6, P_7}\) | the set of $8$ vertices of $C$ | ||||||||||

\(\displaystyle C_1\) | \(=:\) | \(\displaystyle \set {uf, ur, ub, ul, fr, br, bl, fl, df, dr, db, dl}\) | the set of $12$ edges of $C$ | ||||||||||

\(\displaystyle C_3\) | \(=:\) | \(\displaystyle \set {F, B, U, D, L, R}\) | 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 $2.1.7$