The Fibonacci Lattice

The mathematical theory of Penrose tilings gets quite high brow and abstruse, but everything is very simple in one dimension. Then the two shapes are lines of different lengths, which we shall call $A$ and $C$, for Adult and Child (Fibonacci actually studied the dynamics of rabbit populations). Every year each adult has one child and each child becomes an adult. Let us start with a single child

$$C$$ (3.40)

and then repeatedly apply the ``generation rule,''

$$C \mapsto A \quad , A \mapsto AC \quad ,$$ (3.41)

to obtain longer and longer sequences. The first few sequences generated are,

$$C\quad A\quad AC\quad ACA\quad ACAAC\quad ACAACACA.$$ (3.42)

Note the interesting property that each generation is the ``sum'' of the 2 previous generations: $ACAAC$ = $ACA \oplus AC$

In a one dimensional Fibonacci quasicrystal, the longs and shorts could represent the interatomic distances; or the strengths of the bonds between the atoms; or which of two different types of atom is at that position in the chain.