Tetrahedral Symmetry

The only solutions that are obtained in terms of the spherical harmonics for $\lambda\leq10$ are of the third, seventh and ninth order. For each multipolarity we find one independent deformation parameter that characterizes/defines all the other intervening components. We denote those independent parameters $t_3$, $t_7$ and $t_9$. In the lowest order ($\lambda=3$) we find only two related spherical harmonics intervening, viz. the ones with $\lambda=3$ and $\mu=\pm2$:

$$\alpha_{3,\pm2}\equiv t_3.$$ (11)

We find no solutions of order $\lambda=5$. In the $\lambda=7$ order we find four intervening spherical harmonics, i.e., the ones corresponding to $\lambda=7$, $\mu=\pm2$ and $\mu=\pm6$:

$$\alpha_{7,\pm2}\equiv t_7 \quad \textrm{and} \quad \alpha_{7,\pm6}\equiv -\sqrt{\frac{11}{13}} t_7.$$ (12)

Finally, for $\lambda=9$ we obtain

$$\alpha_{9,\pm2}\equiv t_9 \quad \textrm{and} \quad \alpha_{9,\pm6}\equiv +\sqrt{\frac{13}{3}} t_9.$$ (13)

We may conclude that there exist only very few spherical harmonics of order $\lambda\leq10$ that may be used to construct the surfaces of tetrahedral symmetry; but even those that are allowed to intervene are strongly correlated and we have merely 3 independent deformation parameters that characterize the full parametric freedom within tetrahedral symmetry up to the 10$^{th}$ order.