From mathematical order to functional materials: new developments in thermal transport engineering

Abstract Mathematical order such as the golden ratio, Fibonacci sequences, and other quasiperiodic or aperiodic patterns provide a compact design language for functional matter. This review traces their role from describing non-periodic order in quasicrystals, fractals, and optical systems to actively engineering wave and energy transport, with particular emphasis on phononic and thermal phenomena. We highlight periodic superlattices as phononic crystals that control coherent–incoherent crossover, Golomb ruler-based architectures as paradigms of controlled disorder enabling broadband suppression of transport channels, and quasiperiodic sequences such as Fibonacci, Thue–Morse, and Double–Period designs that generate hierarchical scattering and partial localization. Taken together, these advances show how mathematically encoded order can sculpt phonon and photon landscapes in a highly structured way, suggesting mathematics-guided routes toward cross-scale control of heat, light, and related excitations.