## Abstract

The single-scattering properties of the Platonic shapes, namely, the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron, are investigated by use of the finite-difference time-domain method. These Platonic shapes have different extents of asphericity in terms of the ratios of their volumes (or surface areas) to those of their circumscribed spheres. We present the errors associated with four types of spherical equivalence that are defined on the basis of (a) the particle’s geometric dimension (b) equal surface area (*A*), (c) equal volume (*V*), and (d) equal-volume-to-surface-area ratio (*V/A*). Numerical results show that the derivations of the scattering properties of a nonspherical particle from its spherical counterpart depend on the definition of spherical equivalence. For instance, when the Platonic and spherical particles have the same geometric dimension, the phase function for a dodecahedron is more similar than that for an icosahedron to the spherical result even though an icosahedron has more faces than a dodecahedron. However, when the nonspherical and spherical particles have the same volume, the phase function of the icosahedral particle essentially converges to the phase function of the sphere, whereas the result for the dodecahedron is quite different from its spherical counterpart. Furthermore, the present scattering calculation shows that the approximation of a Platonic solid with a sphere based on *V/A* leads to larger errors than the spherical equivalence based on either volume or projected area.

© 2004 Optical Society of America

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