In our recent works [1,2], we analyzed the structural diversity of nanoparticles and the fragmentariness of their structure and formulated the statement that the structural inhomogeneity is a fundamental property of the nanostate. This statement was experimentally confirmed for a number of materials.In particular, reasoning from the results of neutron diffraction and X-ray diffraction investigations of ZrO 2 nanoparticles, Burkhanov et al. [3] proposed a twophase model allowing for the difference between the central and peripheral regions of a particle and their pseudomorphic conjugation. Palosz et al. [4] also used a two-phase model to interpret their diffraction data for SiC, GaN, and diamond nanoparticles. The concept of an "apparent lattice parameter" clearly indicates strong deviations of the position of Bragg peaks (especially at small angles) from the predicted crystallographic positions. The inference was made that a unique lattice parameter for microcrystallites cannot be determined by standard powder diffractometry. The behavior of nanocrystalline powders under pressure cannot be satisfactorily explained in terms of a unique compressibility coefficient. This also counts in favor of the twophase structure of nanoparticles.Earlier [5][6][7], structurally inhomogeneous zirconia nanoparticles that consist of interpenetrating fragments with different symmetries and are characterized by the orientational relationships "incompatible" from the standpoint of classical crystallography were experimentally found for the first time. Regularly oriented fragments with different symmetries, for which the requirements of classical crystallography are not satisfied (the interface need not be a plane, so that the orientational relationships do not necessarily correspond to the Miller indices and the fragments themselves need not be crystals), were referred to as centaurs [5]. The boundaries of fragments are coherent, and, therefore, these particles can be defined as nanostructures with coherent boundaries.Alok Singh and Tsai [8] revealed that the cubic and icosahedral phases in metal alloys can intergrow in a regular oriented manner. It was found that the threefold axes of the cubic phase coincide in direction with the twofold axes of the icosahedral phase and the cubic twofold axes are almost parallel to the icosahedral fivefold axes. The interface is not a plane, and the orientational relationships do not correspond to rational ratios of the corresponding Miller indices and are inconsistent with the assumption that this cubic phase is a quasicrystalline approximant of the icosahedral phase.The structural inhomogeneity of nanoparticles has become evident owing to widespread use of high-resolution electron microscopy [7,9]. Of special interest are materials prepared by ultrafast solidification, because extreme conditions often lead to the formation of vitreous or unstable crystalline structures and quasicrystals. Specifically, it has been demonstrated that fragments of cubic crystals can intergrow to form a hierarchic st...

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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.

The structural inhomogeneity in combination with coherence is characteristic of structures of nanosized particles [1,2]. In this case, the chemical nature of substances (organic, inorganic, biological) appears to be immaterial, which suggests their mutual convergence [3]. The description (and chemical design) of spatially inhomogeneous and hybrid structures should be based on principles that are more general than those accepted in classical crystallography for describing macroscopic crystals by groups of isomorphic mapping of the infinite three-dimensional Euclidean space onto itself. The great diversity of "unusual" structures [4-10] inherent only in nanoobjects can be obtained by mapping (projecting) fragments of high-symmetry structures from different non-Euclidean (in the particular case, projective) spaces onto the three-dimensional Euclidean space E 3 or mapping these fragments onto manifolds embedded in the space E 3 . The concept of projection from one space onto another space with the loss of a part of information can be illustrated by the interrelation between genes and proteins, as well as between the description language and real structures, and is closely related to fundamental problems of "inorganic life" [11].Spatially inhomogeneous structures (centaurs) for which a local short-range order only insignificantly differs from a short-range order of one of the stable (metastable) structural modifications (macroscopic phases) of the material under investigation and different fragments are coherently joined into a unit should exist and be relatively stable in the nanoworld. The requirement for the absence of dangling bonds and substantial disturbances of the mutual coordination of atoms is particularly satisfied for atoms located at interfaces. This implies the absence of interfaces in their usual macroscopic meaning.Specifically, these requirements are met by tetrahedral (diamond-like) structures that, as a whole, have icosahedral symmetry. These structures can exist only in the nanoworld. Let us demonstrate how an icosahedral diamond-like nanoparticle (icosahedral diamond) can be constructed if it is treated as a nanostructure that has coherent boundaries and is composed of (insignificantly distorted) fragments of diamond and lonsdaleite (hexagonal diamond).In our earlier works [12][13][14], we developed the local approach. Within this approach, nanoparticles with coherent boundaries in the general case are assembled from a limited set of building blocks (geometrical structural complexes) determined by the fundamental (specifically, projective) manifolds and the principles of assembling are governed by the topological properties of a fiber space. The geometrical structural complexes of tetrahedrally coordinated structures were derived and it was shown that fragments of crystalline and quasicrystalline structures can be joined together into a unified nanostructure with coherent boundaries [12]. In [15], it was demonstrated that the local approach allows one to explain different types of icosahedral pack...

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