Quasicrystal Diffraction
Quasicrystals exhibit perfect long-range order with rotational symmetries forbidden by classical crystallography — 5-fold, 8-fold, 10-fold, 12-fold. The pattern never repeats, yet its Fourier transform (diffraction pattern) shows sharp Bragg peaks, like a crystal. Dan Shechtman won the 2011 Nobel Prize in Chemistry for their discovery.
f(k) = Σ exp(ik · rj) — structure factor / diffraction amplitude
Classical crystallography's restriction theorem states that only 2-, 3-, 4-, and 6-fold rotational symmetries are compatible with translational periodicity. Five-fold symmetry, like a pentagon, cannot tile a flat plane without gaps or overlaps — yet quasicrystals possess exactly this symmetry, with atoms arranged in an aperiodic but perfectly ordered structure.
This simulation uses the multigrid projection method: superpose n sets of parallel lines at angles 2πk/n. Each polygon in the tiling corresponds to an intersection of lines from two different families. The resulting tiling has perfect n-fold symmetry and is aperiodic — it never repeats, but is not random: local patches reappear infinitely often.
The diffraction pattern of a quasicrystal shows sharp spots — just like a crystal — but arranged with the forbidden symmetry. Each spot can be indexed by n integers (vs. 3 for a 3D crystal), reflecting the higher-dimensional nature: quasicrystals are projections of periodic structures from higher-dimensional spaces (e.g., 5D for icosahedral quasicrystals).
In 1982, Dan Shechtman observed ten-fold diffraction symmetry in rapidly cooled Al-Mn alloy. His paper was initially rejected — Linus Pauling famously said "There are no quasicrystals, only quasi-scientists." After years of controversy, quasicrystals were confirmed and are now known in hundreds of alloys. Shechtman received the 2011 Nobel Prize in Chemistry.