Square-triangle tilings

B. General Properties of Random Square-Triangle Tilings... 

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...

For purposes of illustration, a representative random tiling composed of squares and equilateral triangles (ST tiling) is shown in Fig. 3. This tiling... 

This important theorem refers to two- and three-dimensional structures. It expresses the fact that tiling of the Euclidean plane by regular polygons can be achieved only with the triangle, the square and the hexagon. A four-dimensional periodic structure can allow other symmetry operations. 

Fig. 2.10. Tiling of two-dinnensional Euclidean planes (a) arbitrary lattice, twofold axes 2 (b) rectangles, reflection lines nn (c) diannonds, rectangular centered cell, reflection lines m and glide lines g (d) squares, fourfold axes 4 (e) triangles, threefold axes 3 (f) hexagons, sixfold axes 6, same type of cell as (e)...

---