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We present mathematical theory for understanding the transmission spectra ofheterogeneous materials formed by generalised Fibonacci tilings. Our results,firstly, characterise super band gaps, which are spectral gaps that exist forany periodic approximant of the quasicrystalline material. This theory,secondly, establishes the veracity of these periodic approximants, in the sensethat they faithfully reproduce the main spectral gaps. We characterise superband gaps in terms of a growth condition on the traces of the associatedtransfer matrices. Our theory includes a large family of generalised Fibonaccitilings, including both precious mean and metal mean patterns. We demonstrateour fundamental results through the analysis of three different one-dimensionalwave phenomena: compressional waves in a discrete mass-spring system, axialwaves in structured rods and flexural waves in multi-supported beams. In allthree cases, the theory is shown to give accurate predictions of the super bandgaps, with negligible computational cost and with significantly greaterprecision than previous estimates.

Super band gaps and periodic approximants of generalised Fibonacci tilings