This article presents results of a numerical effort to determine the dielectric polarizabilities of the five regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The polarizability is calculated by solving a surface integral equation, in which the unknown potential is expanded using third-order basis functions. The resulting polarizabilities are accurate to the order of 10 4 . Approximation formulas are given for the polarizabilities as functions of permittivity. Among other results, it is found that the polarizability of a regular polyhedron correlates more strongly with the number of edges than with the number of faces, vertices, or the solid angle seen from a vertex.Index Terms-High-order basis functions, polarizability, polyhedra, surface integral equation.

The polarizability characteristics of nonspherical scatterers, especially cubes and squares, are studied in this paper. A surface integral equation for the electrostatic potential has been numerically solved for a cube and a square-shaped object. Nonuniform gridding of the dielectric object has been used to get a good accuracy in the solution. The results obtained for the polarizability of cubes using the surface integral equation are compared with those obtained from the volume integral equation and show a good agreement. We give simple approximation formulas for the polarizabilities as functions of the permittivity of the inclusions. The results give an improvement compared to earlier literature and also present a way to predict the effective properties of mixtures where cube-shaped inclusions are embedded in dielectric environment, which type of mixtures strongly generalize the effective-medium theories of the present literature.

INTRODUCTIONThe polarizability ␣ is the relationship between the external electric field E and the dipole moment p that is induced in e the inclusion by this field:For a homogeneous dielectric sphere with permittivity ⑀ i embedded in an environment with permittivity ⑀ , the polarize w x ability is a well-known quantity 1, 2 :where V is the volume of the sphere. For ellipsoids, simple w x closed-form formulas for the polarizability can be written 3 . For ellipsoids and scatterers with anisotropic material permittivity, the polarizability is a dyadic.For a scatterer of arbitrary shape, the polarizability cannot be presented in such a simple form. The electrostatic problem has to be solved, where the inclusion is exposed to a uniform field, and this requires, in general, a numerical effort. In an earlier paper, we calculated the polarizability of w x a dielectric cube 4 , where we also discussed previous enuw x merations 5᎐7 of the static polarizing behavior of dielectric and conducting cubes. The aim of this paper is to present the polarizabilities of certain other regular polyhedra, namely, those of a tetrahedron and an octahedron.In general, the polarizability of an arbitrary inclusion is a dyadic. However, due to the symmetry of regular polyhedra, there is no preferred spatial direction, and hence their polarizabilities are multiples of the unit dyadic, in other words, equivalent to scalars.The polarizability is calculated using an integral of the scalar potential over the surface of the scatterer. This potential obeys an integral equation which is solved by a moment method. Due to the difficult field behavior near the edges of the polyhedra, it is advantageous to build a nonuniform grid over the surface. This is shown in more detail in the next section. The potential on the surface of a dielectric object can be w x calculated from the following integral equation 8 :Ѩ nЈ r y rЈ S Ž . In 3 , S is the surface of the object, is the potential of the e incident field, and is the total potential on the surface of Ž . the object. is the ratio of permittivities s ⑀ r⑀ , ⑀ is the i e i permittivity of the inclusion, and nЈ is the outward normal to the surface S at point rЈ.Let the incident field be uniform and polarized in the z-direction. Hence, the incident potential is linear:The dipole moment p can be calculated fromwhere u is a unit normal vector to S. Polarizability ␣ can n Ž . Ž . then be calculated from 1 using the known value from 5 for the dipole moment.Ž . The integral equation 3 is solved using the method of moments. The uniqueness of the solution and convergence of w x the numerical method are discussed in 4 .The faces of the polyhedra are divided into patches whose size diminishes linearly from the center of the face to the edges. This simple method clearly increases the accuracy of w x Ž . the results 4 . Symmetry is used efficiently see Figs. 1 and 2 : the unknown potential values do not need to be calculated everywhere on the surface of the particle. They only have to be calculated on the gray areas show...

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