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Joined: Mon Nov 09, 2015 1:36 am Posts: 460 Location: Ioannina, Greece

We give a solution in the case of the tetrahedron:Definition: A symmetry of a (regular) tetrahedron $S$ is a linear transformation $T:\mathbb{R}^3\longrightarrow\mathbb{R}^3$ with orthogonal matrix which also leaves tetrahedron $S$ unchanged(*), i.e. $T(S)=S$.  Definition: An axis of symmetry of a (regular) tetrahedron is an axis which leaves the tetrahedron unchanged after a rotation of the tetrahedron around that axis, by a certain angle.
Attachment: Lets find the axes of symmetry: The axes $R_1$, $R_2$, $R_3$ and $R_4$ which passes through the vertices $1$, $2$, $3$ and $4$, respectively, and the centroids of the respective opposite faces of the tetrahedron, are axes of symmetry, since they leave it unchanged if it is rotated (clockwise) around one of them by an $\frac{2\pi}{3}$angle, or by an $\frac{4\pi}{3}$angle. The rotation around the $R_1$axis by an $\frac{2\pi}{3}$angle can be represented (described) by the permutation $\rho_1=({2\,3\,4})$. The permutation $\rho_1^2=({2\,4\,3})$ represents the rotation around the $R_1$axis by an $\frac{4\pi}{3}$angle. Similarly we have \begin{align*} \rho_2=({1\,3\,4})\,, \quad \rho_2^2=({1\,4\,3})\,,\quad \rho_2^3=id\\ \rho_3=({1\,2\,4})\,, \quad \rho_3^2=({1\,4\,2})\,,\quad \rho_3^3=id\\ \rho_4=({1\,2\,3})\,, \quad \rho_4^2=({1\,3\,2})\,,\quad \rho_4^3=id\,. \end{align*} The axes $T_1$, $T_2$ and $T_3$ which passes from the midpoints of the respective opposite edges of the tetrahedron, are axes of symmetry, since they leave it unchanged if it is rotated around one of them by an $\pi$angle. We have the permutations \begin{align*} &\tau_1=({1\,2})\,({3\,4})\,,\quad \tau_2=({1\,4})\,({2\,3})\\ &\tau_3=({1\,3})\,({2\,4})\,, \end{align*}which represent (describe) the rotation, by an $\pi$angle, around the $T_1$axis, the $T_2$axis and the $T_3$axis, respectively. Also holds: \begin{align*} \rho_1\,\rho_2=\tau_2\,, \quad \rho_1\,\rho_3=\rho_2\,, \quad \rho_1\,\rho_4=\tau_3\,,\\ \rho_2\,\rho_3=\tau_1\,, \quad \rho_2\,\rho_4=\rho_3\,, \quad \rho_3\,\rho_4=\tau_2\,,\\ \tau_1\,\tau_2=\tau_3\,, \quad \tau_2\,\tau_3=\tau_1\,, \quad \tau_1\,\tau_3=\tau_2\,. \end{align*}The set ${\cal{T}}=\big\langle{\,\rho_{i}\,,\,\tau_{j} \;  \; i=1,2,3,4, \ j=1,2,3}\big\rangle$ equipped with composition of permutations is a group isomorphic to alternating group ${\cal{A}}_4$.
 Definition: A plane of symmetry of a (regular) tetrahedron is a plane which leaves the tetrahedron unchanged after a reflection of the tetrahedron regarding that plane.
Attachment: Similarly the reflections regarding a plane of symmetry can be represented as permutations.
In total all the above permutations form a group isomorphic to symmetric group ${\cal{S}}_4$. (*) Leaves the tetrahedron unchanged, in the sence that we have the same picture of it, without distinguishing different faces, different edges or different vertices at first.
_________________ Grigorios Kostakos

