Hexagonal tiling

The triangular and hexagonal tilings are dual to each other, while the square tiling is self-dual. For other parameters (r, q), the tiling r, q lives in hyperbolic space H2 (see many pictures in [EpsOO]) but we will not need it. 

Fig. 63 X-shaped quaternary bolaamphiphile 189 forming hexagonal cylinder phases with models showing (a) the three-color, (b) two-color and (c) single color hexagon tiling patterns in the left bottom comer the core-shell structure of the mixed cells is illustrated dark = Rp-chains white = Rsi-chains gray = mixed cells [42]...

Fig. 14 Interacting triangular tiles, (a) Sequences are designed so to assemble in two different triangular tiles with complementary sticky ends, (b) Six connected triangles form hexagonal units, which (c) create a two-dimensional crystal, (d) AFM image (edge size is 133 nm) of two overlapped hexagonal tilings. Adapted with permission from [69]...

Researchers used an STM to obtain an image of the HB-DC monolayer. The image essentially matched what they had expected HB-DC molecules were arranged in a crystal-like pattern in which every molecule was held in place by other molecules surrounding it. Because of their shape, the molecules looked like a collection of hexagonal tiles neatly laid out on the copper surface. 

Fig. 10 Complex DNA motifs, a DX triangle self-assembly to a pseudohexagonal lattice [57], b DNA triangles ligated to produce a linear array [58], c DNA three-point star motif assembly to the hexagonal arrays [59], d ffexagonal structure composed of six triangular complexes, and extended to a pair of overlapping hexagonal tilings [60]. e 16 cross-tiles construct directly to one square [62]. f Self-assembly of the cross motifs to 2D lattice [61], Reproduced with permission from cited references...

A further orthorhombic member of the -phase family is the -phase. It has lattice parameters a — 1.9902 nm, b — 1.6612 nm and c = 1.4460 nm and can coexist with the other orthorhombic -phases taking well-defined orientation relationships [28]. As for 6, the structure of the -phase can be described by Mackay-type clusters on the vertices of a lattice of flattened hexagons. However, while flattened hexagons in two alternating orientations are used for the Eg-phase, for the -phase only hexagon tiles of a single orientation are used, which are arranged in parallel (see Section 2.3). 

Fig. 3. (a) High-resolution transmission electron micrograph of Sj-Al-Pd-Mn along the [010] direction with superposed hexagon tiling, (b) Hexagon tiling. 

There exists a one-to-one correspondence between the tiling description and the atomic positions. Similarly to a unit cell, each the possesses a defined atomic decoration. Fig. 5 shows the decoration of the hexagon tile for the F, I, and P layers. The corresponding tiles for the /, P, Pi, Pj, and P layers are related to those shown through the respective symmetry operations (Section 2.2). The corresponding alternately oriented hexagon is obtained via clockwise rotation by 72°. 

Preservation of local order and coordination as in the ideal structure. In each layer the structure, and accordingly the decoration of the hexagon tiles, displays certain preferred local arrangements. This particularly concerns the vertices of the hexagon lattice, i.e. the corners of the hexagon tiles, which correspond to [010] columns of Mackay-type clusters. 

Despite the (0 01) mirror symmetry of the tUes representing the phason hne, its atomic decoration is not fuUy symmetric. Small deviations occur for some A1 positions in the P layer, which are due to the required compatibility with the hexagon tiles. However, according to the structure model [24] the hexagon tiles in the P layer by themselves are not symmetric with respect to their long axis (Fig. 5) and the corresponding atom positions exhibit low occupancy factors. 

As described in Section 2.3, the hexagon tiling is constructed by drawing lines connecting neighboring cluster centers. This stiU holds for each individual position of the phason line shown in Figs 9(a)-9(d). However, this does not imply that a vertex jump corresponds to the movement of a complete cluster from the initial to... 

Related phases and phase transitions in terms of phason planes In Section 2.2, we have introduced the fact that the ss-phase is the basic structure of a family of related phases, the s-phase family. Fig. 4 shows examples of corresponding hexagon tilings. The S28-phase [Fig. 4(a)] is represented by a tiling which consists of hexagons and banana pentagons. The latter are thus structural elements of the S28-phase, while they are defects in the ss-phase. 

Gays are an essential component of soils, to which we owe our survival, and they are also the raw materials for some of mankind s most ancient and essential artefacts pottery, bricks, tiles, etc. Clays are formed by the weathering and decomposition of igneous rocks and occur typically as very fine panicles e.g. kaolinite is formed as hexagonal plates of edge. 1-3 p m by the weathering of alkaline feldspar... 

Each graphene sheet is composed of rings of carbon atoms arranged on a hexagonal 2D tiling. This form of carbon has n bonds in addition to the a bonds, as shown in Fig. 4.6, leading to a bond order of 1.33. 

Figure 4.2 An Archimedean tile morphology for blends of poly(2-vinylpyridine-/)-isoprene-/)-vinylpyridine) with poly(styrene-/)-4-hydroxystyrene) in a 2 1 vinylpyridine/hydroxystyrene blend. The vinylpyridine/hydroxystyrene domains are the cylinders at the vertices of polystyrene hexagons within a polyisoprene continuous phase. Reprinted from Asari et al. (2006). Copyright 2006 American Chemical Society.

In general, the cadmium halides show in their crystal structure the relation between polarizing effect and si/e of anion. The tluoride has tile smallest and least polarizable anion of Ihe lour and forms a cubic structure, while the mure polarizable heavy halides have hexagonal layer structures, increasingly covalent and al increasing distances apart in inxler down tire periodic table, in solution the halides exhibit anomalous thermal and transport properties, due primarily to the presence of complex ions, such as CDlr and CdBr r. especially in concentrated solutions or those containing excess halide ions. 

---