of the group $O(3)$ of symmetries of the sphere. The subgroups of interest are $O(2)$, $D_k$, and the exceptional subgroups $T$, $O$, and $I$. $O(2)$ consists of the symmetries of a circle; patterns which are axisymmetric (for some orientation) are in this category. $D_k$ describes the symmetries of a $k$-gon, and thus patterns which are invariant under reflection and rotation by $2\pi/k$ (about a fixed axis). The three exceptional subgroups are associated with the five Platonic solids: $T$ (tetrahedron), $O$ (octahedron or cube), and $I$ (icosahedron or dodecahedron).

The Platonic solids are regular polyhedra. Although a drop does not have angular vertices and flat faces, a polyhedron with the same symmetry properties can be constructed from a drop in a $T$, $O$, or $I$ configuration by assigning a local maximum on the surface to a vertex and a local minimum to a face. The dual of a polyhedron is obtained by inverting its vertices with its faces, an operation which preserves symmetry. Similarly, the dual of a drop can be formed by inverting maxima and minima. An oscillating drop provides an ideal opportunity to observe duality: since a location which is the site of a maximum contains a minimum after 