Triangular tiling

The icosahedron is a roughly spherical object that is built up from 20 identical equilateral triangles. These triangular tiles are arranged side by side in such a way that they enclose the volume inside the icosahedron. Figure 16.3 shows a view of an icosahedron where there are 5 tiles at the top, 5 at the bottom, and 10 in a band around the middle region. These faces are identical in other words, identical icosahedra are obtained irrespective of which of the vertices is at the top in Figure 16.3a. 

Figure 16.S Schematic illustration of the way the 60 protein subunits are arranged around the shell of safellite tobacco necrosis virus. Each subunit is shown as an asymmetric A. The view is along one of the threefold axes, as in Figure 16.3a. (a) Three subunifs are positioned on one triangular tile of an Icosahedron, in a similar way to that shown in 16.4a. The red lines represent a different way to divide the surface of the icosahedron into 60 asymmetric units. This representation will be used in the following diagrams because it is easier to see the symmetry relations when there are more than 60 subunits in the shells, (b) All subunits are shown on the surface of the virus, seen in the same orientation as 16.4a. The shell has been subdivided into 60 asymmetric units by the red lines. When the corners are joined to the center of the virus, the particle is divided into 60 triangular wedges, each comprising an asymmetric unit of the virus. In satellite tobacco necrosis virus each such unit contains one polypeptide chain...

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]...

While the pentagon does not tile the plane, it does indeed tile the sphere. Twelve pentagons make up ti e pentagonal dodecahedron whose vertices lie on the sphere. Similarly, the icosahedron is a triangular tiling of the sphere with fivefold rotational symmetry. 

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. 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...

Figure 3-7. Some examples of two dimensional cluster structures built by tiling triangular, or triangular and square fragments. The structures of...

A type of tile construction. The ceramic wear plate has a triangular spigot on its back face mating with a triangular recess on the front face of the backing plate to which it is bonded or brazed. II provides support and extra safety should the bond or braze fall... 

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