Fibonacci sequence/chain

This might be a reason for the stability of quasiperi-odic systems where r plays a role. A prominent 1-D example is the Fibonacci sequence, an aperiodic chain of short and long segments S and L with lengths S and L, where the relations L S=z and L + S = zL hold. A Fibonacci chain can be constructed by the simple substitution or inflation rule L LS and S L (Table 1.3-6, Fig. 1.3-10). Materials quasiperiodically modulated in 1-D along one direction may occur. Again, their structures are readily described using the superspace formalism as above. 

Fig. 1.3-10 1 -D Fibonacci sequence. Moving downwards corresponds to an inflation of the self-similar chains, and moving upwards corresponds to a deflation...

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. 

Figure 18.1 liiustration of the generation of a Fibonacci chain of objects (a) or a sequence... 

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