Prologue

Summary

The theory of harmonic functions was initiated in 1782 by Laplace, when he derived the partial differential equation that is known today as Laplace's equation. The same year Legendre developed the theory of zonal spherical harmonics, which is a solution of the Laplace equation with axial symmetry, while Laplace himself solved his equation in spherical geometry without any symmetry, introducing the concept of tesseral spherical harmonics. Both papers were published in 1785 [230, 233].

The sphere is invariant under rotation and therefore provides the geometrical visualization of isotropy. In an anisotropic space however, where only a finite number of symmetries are possible, the sphere is transformed into an ellipsoid. The study of harmonic functions in the presence of anisotropic structure, which is undertaken in the present book, is more complicated by far than the corresponding study of harmonic functions in the presence of isotropy. The ellipsoidal shape appears naturally in many different forms. For example, Rayleigh has proved that the ultimate shape of pebbles, as they are worn down by attrition, is a generic ellipsoid, see [36, 124, 126, 128, 129, 288]. It is also known that the RGB points, which determine the color of objects in our visual neuronal system, exhibit color insensitivity whenever they vary in a small ellipsoid [184]. Many more cases appear in physics, such as the inertia ellipsoid in mechanics, the directivity ellipsoid, the reciprocal ellipsoid in wave propagation within crystalographic structures, and so on.