Nonconnected atomic surfaces for quasicrystalline sphere packings

Fractals of 3D stars with icosahedral symmetry

The method for constructing 3D fractals of stars with icosahedral symmetry is proposed. A great stellated dodecahedron (I) and a small stellated dodecahedron (D) were selected as building elements. The faces of the polyhedra are equal in size. The initial polyhedron (I or D) is replicated and the copies are placed so that its centres coincide with the vertices of the star (I or D), which is not displayed. This star is called by the author the generalized star, its size is τ ^N times the size of the initial star, where τ  =  (1  +  5^0.5)/2  ≈  1.618 is the golden mean, N is a non-negative integer. At each next step of the construction, the previous prefractal is replicated and the copies are placed so that its centres coincide with the vertices of the generalized star. The series of integers N forms a non-decreasing sequence. The types of generalized stars (I or D) can vary at every step. Coinciding points are counted once. Initial polyhedra can touch each other by vertices, overlap each other or stand separately. Like in the Penrose 3D tiling, all the vertices of these fractals belong to the symmetric projection of a simple 6D cubic lattice onto a 3D space.

Dodecagonal tiling in mesoporous silica

Recent advances in the fabrication of quasicrystals in soft matter systems have increased the length scales for quasicrystals into the mesoscale range (20 to 500 ångströms). Thus far, dendritic liquid crystals, ABC-star polymers, colloids and inorganic nanoparticles have been reported to yield quasicrystals. These quasicrystals offer larger length scales than intermetallic quasicrystals (a few ångströms), thus potentially leading to optical applications through the realization of a complete photonic bandgap induced via multiple scattering of light waves in virtually all directions. However, the materials remain far from structurally ideal, in contrast to their intermetallic counterparts, and fine control over the structure through a self-organization process has yet to be attained. Here we use the well-established self-assembly of surfactant micelles to produce a new class of mesoporous silicas, which exhibit 12-fold (dodecagonal) symmetry in both electron diffraction and morphology. Each particle reveals, in the 12-fold cross-section, an analogue of dodecagonal quasicrystals in the centre surrounded by 12 fans of crystalline domains in the peripheral part. The quasicrystallinity has been verified by selected-area electron diffraction and quantitative phason strain analyses on transmission electron microscope images obtained from the central region. We argue that the structure forms through a non-equilibrium growth process, wherein the competition between different micellar configurations has a central role in tuning the structure. A simple theoretical model successfully reproduces the observed features and thus establishes a link between the formation process and the resulting structure.

Connecting icosahedral and dodecagonal order

A mapping is presented which transforms a flat square-triangle tiling with dodecagonal symmetry into a corrugated layer of tiles, whose edges are oriented parallel to the symmetry axes of an icosahedron. This can be considered as an intermediate state between the icosahedral and the dodecagonal one.