Quasicrystals have long-range order without periodicity. Their diffraction patterns show sharp Bragg peaks with forbidden symmetries — five-fold, ten-fold, twelve-fold — that no periodic lattice can produce. The atomic positions follow deterministic rules, but the rules don't repeat. Computing electronic structure for these materials has required either approximating the quasicrystal as a large periodic crystal (sacrificing the aperiodic order) or treating it as disordered (sacrificing the long-range correlations).
Nop, Smith, Koschny, and Paudyal take the mathematically natural approach: solve the Schrödinger equation in the higher-dimensional periodic space that the quasicrystal projects from. Every quasicrystal can be described as a “cut-and-project” — a slice through a periodic lattice in a higher-dimensional space, where the slice is tilted at an irrational angle so the resulting pattern in the physical lower-dimensional space is aperiodic. The periodicity lives in the higher-dimensional parent; the aperiodicity is the shadow.
Their density functional theory formulation works in the parent space, where translational symmetry holds and Bloch's theorem applies. The electronic states are computed as periodic functions in the higher-dimensional space, then projected to the physical quasicrystal. The projection preserves the physical observables — electron density, total energy, band structure — while the computation benefits from the periodicity that the physical system lacks.
The structural insight: the quasicrystal's electronic states are actually periodic — in a space you can't directly access. The aperiodicity is an artifact of projection, not of the underlying physics. Computing in the parent space treats the material as what it mathematically is, rather than what it appears to be.