Let us demonstrate how an icosahedral diamond-like nanoparticle can be constructed if it is treated as a nanostructure with coherent boundaries and is composed of insignificantly distorted fragments of diamond and lonsdaleite. The geometric principles used for assembling such structures are based on the local approach (Shevchenko

*et al.*, 2004

, 2005

). Within this approach, nanoparticles with coherent boundaries in the general case are assembled from a limited set of building blocks determined by the fundamental manifolds and the principles of assembling are governed by the topological properties of a fibre bundle. The great diversity of ‘unusual’ structures can be obtained by mapping or projecting fragments of highly symmetrical structures from different non-Euclidean spaces into the three-dimensional Euclidean space or mapping these fragments onto curved manifolds embedded into the Euclidean space. In particular, the substructures of polytopes,

*i.e.* regular tilings of the three-dimensional space with positive curvature, are of special interest. They may be considered as regular tilings of the three-dimensional Riemannian space or (being embedded into four-dimensional Euclidean space) as four-dimensional analogues of Plateau polyhedra or as abstract group manifolds with corresponding Coxeter groups (Coxeter, 1973

).

An icosahedron can be assembled by joining 20 slightly distorted regular tetrahedra face to face. This simplest consideration leads to the classical concept of multiple twinning widely used when describing the diamond-like structures (Bühler & Prior, 2000

; Shenderova

*et al.*, 2003

; Son & Chung, 2004

). The Euclidean space may not be filled by ideal tetrahedra or icosahedra but there exists the closest packing of ideal tetrahedra in the curved space – namely the {3,3,5} polytope. Cutting various fragments from it and projecting them with slight distortions into the Euclidean space, one can obtain a lot of finite packings with almost icosahedral arrangement of atoms everywhere (

*e.g.* Sadoc & Mosseri, 1982

; Kléman, 1989

; Lord & Ranganathan, 2001

; Lord

*et al.*, 2006

). In the general case, each tetrahedron may suffer unique distortions during this procedure, so that the true icosahedral symmetry of the packing as a whole may be hidden or lost. The formation of an icosahedral packing is not at all an accidental cyclic twinning of nearly suitable building blocks. Only simplest icosahedral structures may be considered as cyclic multiple twins. We are sure as well that it is not enough for shells to have suitable sizes like Russian dolls to build up the regular multishell structures. In the general case, there exist rigorous assembling rules, which are strictly predefined by the true group-theoretical equivalence of corresponding blocks in the parent structure, which may be not realizable in the Euclidean space but does exist in a certain curved (or projective) space.

As a special case, an icosahedron built up of 20 tetrahedra may be obtained as a result of the cut-and-project procedure by choosing the packing origin in one of the vertices of polytope {3,3,5} and confining oneself to cutting no more than nearest neighbours. But the abilities of the approach will not rest by designing just another ordinary multiple twin. For instance, the rod-like and zigzag structures spatially compatible with interpenetrating multilayer icosahedral packings may be designed by projecting the polytope fragments onto the Clifford surfaces. As known, the Clifford surfaces are special kinds of manifolds of zero Gaussian curvature defined in the elliptical space. Corresponding surfaces in the Euclidean space are cylinders. Thus, the structures of multilayer nanotubes and ‘magic’ nanowires may be designed.

New packings with nearly icosahedral motif may be derived using the generalized cut-and-project procedure. First, one can increase the number of shells in the packing up to the equatorial cut-off of the polytope. Second, one can shift the origin of the structure being projected from vertex towards face or cell centre in the curved space. Third, one can perform the geodesic design of highly symmetrical structure in the curved space and only afterwards apply the cut-and-project procedure. Besides, one can increase the complexity of the non-Euclidean packing by taking multiple copies of the polytope in the same way as face-centred or body-centred cubic lattices may be derived from a simple cubic lattice in the Euclidean space.

Another way is to fill the fundamental regions of the non-Euclidean structure by slightly distorted multiple copies of some Euclidean packing which leads to the ‘decorated’ polytope. For example, every 120 tetrahedra in the structure of polytope {3,3,5} may be considered in curved space as a full analogue of the unit cell of a crystal. Then the tetrahedrally shaped fragments of some cubic crystal lattice may be cut out and projected into the unit tetrahedron of the polytope {3,3,5} so that the face-to-face boundaries between nearest ‘decorated’ tetrahedra would reproduce the atomic structure of the Σ3 twin boundaries in cubic crystals. In the curved space, the tetrahedral fragments should be multiplied by the generators of the Coxeter group [3,3,5]. After that, the fragments of the decorated polytope should be projected back into the Euclidean space resulting in a family of complex icosahedral diamond-like shell packings.

Constructed in such a way, diamond nanoparticles have a shell structure and a nearly spherical shape. In the simplest case (Shevchenko & Madison, 2006

*a*
), each shell contains 20

*k*
^{2} atoms (20, 80, 180, 320, 500,…), and the particle as a whole consists of 20

*k*(

*k* + 1)(2

*k* + 1)/6 atoms (20, 100, 280, 600, 1100,…), where

*k* is the total number of shells. Special cases of this ‘magic’ series are provided by endohedral nanodrops of the (H

_{2}O)

_{100} water, which were found in cavities of giant oxomolybdate clusters (Müller

*et al.*, 2003

), and the (H

_{2}O)

_{280} Dzugutov clusters (Doye

*et al.*, 2001

). These numbers are also characteristic of carbon onions (Terrones

*et al.*, 2003

). Icosahedral diamond-like particles can undergo reversible transformation into onions by shell smoothing and backward puckering without any jumps of atoms between shells or migrations of atoms within intershell spacings. Let us illustrate this.

Fig. 1 shows the consecutive shells of an example of the icosahedral diamond-like nanoparticle. Its structure may be described either in terms of building units (Shevchenko & Madison, 2006

*a*
) or in terms of closed shells (Shevchenko & Madison, 2006

*b*
). In terms of building units, the core of this particle consists of 20 atoms forming a regular dodecahedron. Columns of ‘barrels’ are attached to each of its faces. The number of barrels in each column is defined by the frequency parameter of the geodesic design of the polytope. In general, each column should be terminated by dodecahedra from both sides. Each dodecahedron in the structure of an arbitrary icosahedral diamond-like nanoparticle corresponds to a certain vertex of the polytope {3,3,5}, whereas columns of barrels correspond to its edges. Dodecahedra and barrels form a scaffold. The remaining space should be regularly filled with fragments of diamond and lonsdaleite. The lonsdaleite fragments fill the surfaces corresponding to the faces of the polytope in the curved space. The diamond fragments fill the volume inside the tetrahedra corresponding to the cells of the polytope. In the simplest case, there is only one central dodecahedron and twelve columns of barrels in an icosahedral diamond-like nanoparticle (Shevchenko & Madison, 2006

*a*
). Another example has been reported elsewhere (Zeger & Kaxiras, 1993

).

A generalized cut-and-project procedure makes it possible to design a lot of structures characterized by almost perfectly tetrahedral bonding of atoms and icosahedral point symmetry. The analogous structures with tetrahedral or dihedral point symmetry and almost icosahedral motif are possible, too. Similarly, related structures of multishell nanowires with hidden icosahedral motif may be designed. An example of the diamond-like helical structure has been presented recently (Shevchenko & Madison, 2006

*a*
). On the one hand, the arrangement of nearest neighbours in that structure is almost like that in a diamond crystal and, on the other hand, it has the overall symmetry characteristic of the Boerdijk–Coxeter helix. All these structures (multishell spherical particles and wires) are structurally compatible. It means they can interpenetrate each other and may be joined together coherently according to the rules of the parent structure existing in the curved space. Let us emphasize that such complex interpenetrating coherent nanosized structures are not twins in the classical meaning of this word.

In terms of closed shells, there are puckered and smoothed shells. Inserting the puckered consecutive shells (Fig. 1, left) into another one obtains the diamond-like particle. The same shells but smoothed (Fig. 1, right) form nested fullerene–onion. Both sets of shells are equal topologically and may be reversibly transformed by smoothing and puckering of atomic networks. It should be emphasized that, in the framework of the mechanism under consideration, the transformation of an icosahedral diamond-like nanoparticle into a shell nanoparticle is not accompanied by breaking of any chemical bonds and leads only to a change in their character and direction. In terms of atomic orbitals, the reversible structural transformations in carbon nanoparticles correspond to dehybridization and rehybridization of the bonds. The topological integrity of the network as a whole remains unchanged.

Another possibility is the coherent coexistence of diamond-like core with onion-like shells. Let us suppose that outer shells undergo the smoothing transformation, whereas the inner core remains diamond-like. This case corresponds to the composite icosahedral core-shell particle without any grain boundaries (in the classical sense) between diamond-like and graphite-like fragments. It is the special type of ordering specific for the nanostate, which cannot be explained in classical terms of twinning or polysyntaxy.

Thus, the structures of icosahedral diamond-like nanoparticles can simultaneously involve fragments with specific features of diamond, lonsdaleite, graphite and carbon onions joined coherently. They can serve as a good model accounting for the structure of detonation nanodiamonds, as well as for the structural transformations of nanodiamonds into onion-like carbon structures and

*vice versa* (Aleksenskii

*et al.*, 1999

; Banhart

*et al.*, 1997

; Tomita

*et al.*, 2000

, 2002

; Roddatis

*et al.*, 2002

; Ponomareva & Chernozatonskii, 2003

; Banhart, 2004

; Mykhaylyk

*et al.*, 2005

).