  The symmetric group $S_6$ that permutes the six five-fold axes of an
icosahedron is introduced to go beyond the simple rotations that constitute the
icosahedral group $I$. Owing to the correspondence $h\leftrightarrow d$, the
calculation of the Coulomb energies for the icosahedral configurations $h^N$
based on the sequence $O(5) \supset S_6 \supset S_5 \supset I$ can be brought
to bear on Racah's classic theory for the atomic d shell based on $SO(5)
\supset SO_L(3) \supset I$. Among the elements of $S_6$ is the kaleidoscope
operator ${\cal K}$ that rotates the weight space of SO(5) by $\pi/2$. Its use
explains some puzzling degeneracies in d^3 involving the spectroscopic terms
^2P, ^2F, ^2G and ^2H.
