Small ice crystals have been found to occur naturally in high concentrations at very cold temperatures, such as in cirrus located near the tropopause and polar stratospheric clouds. The scattering properties of these small crystals are important to space-borne remote sensing, in particular, the retrieval of cirrus properties based on near-infrared and infrared channels. It is a common practice to apply a spherical approximation to the calculation of the optical properties of these ice crystals. In this study we first investigate the error of using a spherical approximation by comparison of the optical properties of Platonic solids and their spherical equivalences. The Platonic solids, named in honor of the famous ancient Greek philosopher Plato, include five geometric shapes: tetrahedron, cube, octahedron, dodecahedron, and icosahedron, which have 4, 6, 8, 12, and 20 surface faces, respectively. We show that the commonly used spherical approximation based on the equivalence of the ratio of particle volume to particle projected area is worse than the spherical approximation in terms of surface area or volume, depending on specific particle geometry. Based on observational evidence, we suggest that droxtals more accurately represent the shape of these small ice crystals. The single-scattering properties of ice droxtals have been computed at visible and infrared wavelengths using the finite-difference time domain method for size parameters smaller than 20. Further study of the optical properties of larger droxtals (size parameter > 20) will be carried out using the geometric optics method.