The classical singularities of elliptically polarized light are points of circular (linear) polarization, characterized by a half-integer (integer) topological index. On average, in any plane of a random ellipse field there is of the order of one each of these classical singularities per coherence area. It is shown that every ellipse in such a field is a multiple singularity characterized by nine different topological indices: Three indices characterize rotations of the principal axis system of the surrounding ellipses, and six indices characterize a one- or two-turn spiral precession of these axes. The nine indices can divide the field into 32, 768 different volumes with different structures separated by singular surfaces (grain boundaries) on which an index becomes undefined. This unprecedented proliferation of singularities and structures can occur in other three-dimensional systems in which individual elements are described by unique principal axis systems, for example, liquid crystals, and should be sought in such systems.
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