It is demonstrated that the structural inhomogeneity of the nanostate is a fundamental property and can be adequately explained in terms of the algebraic geometry when the four-dimensional fiber space is chosen as a hypothetical praphase of a nanoparticle. Zirconia nanoparticles ZrO 2 with coherent boundaries between their constituent fragments are treated as cross sections of this praphase by three-dimensional Euclidean hyperplanes. The monoclinic, tetragonal, and orthorhombic zirconia structures are assembled from the capped octahedra Z 7 and the Bernal polyhedra Z 8 and Z 9 that are geometrical structural complexes (building blocks) of fluorite-like structures. The interrelated constructions of finite projective geometries are determined. These constructions make it possible to specify graphs of the Z 7, Z 8, and Z 9 polyhedra and to simulate the corresponding ZrO 2 phases as fiber bundles associated with one principal fiber bundle, namely, the ZrO 2 praphase. A priori possible mutual transformations in zirconia are considered, and new structural forms of nanoparticles assembled from the Z 7, Z 8, and Z 9 polyhedra are predicted.
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