Aperiodic tiling

One way of explaining the nature of atomic order in QCs is simply to say that order is dictated by a rule other than periodicity, as it is in a nonperiodic mathematical construction. (Actually, there are special constraints on the type of mathematical construction that can produce quasiperiodicity [13].) In discussions of QCs, two types of constructions are commonly invoked the Fibonacci sequence or chain, and aperiodic tilings. 

A non-periodic (aperiodic) tesselation or tiling is one that cannot be created by the repeated... 

Figure 3.17 Kites and darts (a) a rhombus with each edge equal to the golden ratio, GR (b) a kite, with arcs that force an aperiodic Penrose tiling when matched (c) a dart, with arcs that force an aperiodic Penrose tiling when matched (d) five kites make up a decagon...

Quasi-crystals have macroscopic symmetries which are incompatible with a crystal lattice (Section 2.4.1). The first example was discovered in 1984 when the alloy AlMn is rapidly quenched, it forms quasi-crystals of icosahedral symmetry (Section 2.5.6). It is generally accepted that the structure of quasicrystals is derived from aperiodic space filling by several types of unit cell rather than one unique cell. In two-dimensional space, the best-known example is that of Penrose tiling. It is made up of two types of rhombus and has fivefold symmetry. We assume that the icosahedral structure of AlMn is derived from a three-dimensional stacking analogous to Penrose tiling. As is the case for incommensurate crystals, quasi-crystals can be described by perfectly periodic lattices in spaces of dimension higher than three in the case of AlMn, we require six-dimensional space. 

The key goal of this system is to produce periodic (or aperiodic) assays from DNA motifs that can act as tiling elements in two or three dimensions. These tiles must be rigid components, because flexibility can lead to un-... 

The second mathematical construction is the aperiodic tiHng. The example shown in Figure 18.2 is the Penrose P3 tiling [14]. Here, the two rhombi form the basis of the construction. Penrose reaHzed that if the two rhombi are assembled according to specific matching mles, they can cover the entire two-dimensional plane [15]. They form a pattern that can be considered a kind of two-dimensional extension... 

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