This Demonstration shows the spectral properties of the Fibonacci quasicrystal from both algebraic and graphical points of view; in the latter case in real and reciprocal space. The Fibonacci quasicrystal is the most studied one-dimensional (1D) quasiperiodic model structure, given its interesting applications in photonics and acoustics.

Like all other 1D quasiperiodic structures discussed in [1], the construction of such a system is based on a substitution sequence defined by a two-letter alphabet

and a substitution rule

,

that can be expressed in matrix form as

,

using the substitution matrix

.

This procedure can be iterated

times; it corresponds to the generation of a single periodic approximant for the 1D quasicrystal system, which can be replicated

times to give an adequate approximation (patch) for an infinite quasicrystal [2].

This Demonstration implements the two-letter word generation based on such a rule up to a finite number

of nesting substitutions using the appropriate controls. For the Fibonacci substitution matrix, the computed eigenvalues are the golden ratio

and its negative reciprocal

.

It can be easily checked that at each nesting iteration the frequencies of the letters

and

are consecutive Fibonacci numbers. As

tends to infinity, the ratio of the frequencies of

to

tends to

, since the ratio of two consecutive Fibonacci numbers tends to the golden ratio

.

The frequencies of the two letters

and

at each nesting iteration

have been computed in two reciprocally consistent ways, by simply counting the number of occurrences of the letters

and

in the generated letter sequence and by implementing the following power matrix relation:

.

By selecting the "plots" setter, you can see the resulting 1D spectrum in real space for a single periodic approximant (impedance plot), where

and

are assigned to spacings of lengths

and 1, respectively. The distance from the origin

is normalized by the period of the letter sequence (or periodic average structure)

, where

is set to 1, in accordance with [1]. A bar of unit length 1 is put at each

coordinate point in order to give a pictorial idea of the quasiperiodicity of the system.

You can visualize the reciprocal space representation in Fourier space by selecting either the "Fourier" setter for the corresponding transformed power spectrum or the "transmission" setter for the corresponding transmittance spectrum on a logarithmic scale. The slider control "number of approximants

" lets you obtain a denser sampling of reciprocal space points

in order to see higher-order peaks in the Fourier spectrum. This is achieved numerically by replicating the fundamental letter sequence used to construct the single periodic approximant

times. The computed Fourier power spectrum can also be considered as a simulation of the diffraction pattern of the 1D Fibonacci quasicrystal.