The Quintic Origin of Aperiodicity

Why is the ground state of a quasicrystal aperiodic? Recent quantum-mechanical simulations have demonstrated that certain quasicrystalline alloys have lower energy than any periodic alternative. BEC experiments have stabilised octagonal, decagonal, and dodecagonal quasicrystal phases at zero temperature with five collective excitation modes each, all exhibiting aperiodic structure. Neither result explains the mechanism. We propose that the mechanism is algebraic. Five-fold symmetry in three dimensions requires the golden ratio φ = (1+√5)/2 as a geometric constant. φ is irrational and therefore incompatible with periodic lattices, which require rational basis-vector ratios. What distinguishes φ from other…