Diffraction patterns
Compute the 2D diffraction pattern of point arrangements — the Fourier transform that reveals hidden symmetry. Select a lattice or quasicrystal and watch its reciprocal-space structure appear. The centerpiece: Penrose tiling vertices producing the “impossible” 10-fold symmetry that Dan Shechtman observed in 1982.
I(kₓ, kᵧ) = |Σⱼ exp(−i(kₓ·xⱼ + kᵧ·yⱼ))|² → power spectrum of point positions → reciprocal space
What is diffraction?
When waves — X-rays, electrons, neutrons — scatter from an arrangement of atoms, they interfere constructively and destructively. The resulting diffraction pattern is the Fourier transform of the atomic positions: it reveals the structure's hidden symmetry and periodicity without requiring you to see the atoms directly. X-ray diffraction gave us the structure of DNA (Franklin, Watson, Crick, 1953), the atomic arrangement of every known crystal, and ultimately the field of structural biology. The bright spots in a diffraction pattern correspond to the spatial frequencies present in the arrangement — the sharper and more ordered the arrangement, the sharper the peaks.
The crystallographic restriction
A fundamental theorem of geometry: in two dimensions, only 2-fold, 3-fold, 4-fold, and 6-fold rotational symmetries are compatible with translational periodicity (tiling the plane by repeating a unit cell). 5-fold, 7-fold, 8-fold, and higher symmetries are forbidden — you cannot tile the plane periodically with pentagons. This was taken as settled mathematical law. Any diffraction pattern showing 5-fold or 10-fold symmetry was simply impossible — or so everyone believed.
Shechtman's discovery
On April 8, 1982, Dan Shechtman at the National Bureau of Standards was studying a rapidly cooled aluminum-manganese alloy. When he examined the electron diffraction pattern, he saw something that should not exist: sharp Bragg peaks arranged with 10-fold symmetry. His notebook entry read “10-fold???”. The pattern was unmistakable — bright, sharp, well-defined spots, not the diffuse ring of an amorphous material, but arranged with a symmetry that the crystallographic restriction theorem explicitly forbade. He was ridiculed. His research group leader asked him to leave the group. The double Nobel laureate Linus Pauling publicly declared, “There is no such thing as quasicrystals, only quasi-scientists.” Shechtman persisted. He was awarded the 2011 Nobel Prize in Chemistry, alone.
Why quasicrystals work
The resolution of the paradox: quasicrystals have long-range order (producing sharp diffraction peaks) but no translational periodicity (no repeating unit cell). The crystallographic restriction only applies to periodic structures. Quasicrystals are built from two or more tile types arranged according to matching rules that enforce order without repetition. The ratio between their characteristic lengths is typically the golden ratio φ = (1 + √5)/2 — an irrational number that guarantees aperiodicity. The Penrose tiling, independently discovered by Roger Penrose in 1974, is the canonical example: two rhombus shapes that tile the plane aperiodically with local 5-fold symmetry.
Sharp peaks without periodicity
This simulator lets you see the key insight directly. Switch between the square lattice (sharp peaks, 4-fold symmetry — conventional crystal) and the Penrose tiling (sharp peaks, 10-fold symmetry — quasicrystal). Both produce sharp diffraction peaks, meaning both have long-range order. But the Penrose pattern has a symmetry that no periodic lattice can possess. The random point preset, by contrast, produces only a diffuse ring — no order at all. This is the conceptual revolution: long-range order does not require periodicity. Order and periodicity, conflated for centuries, are genuinely different things.
2025: quasicrystals are ground states
For decades, a question lingered: are quasicrystals thermodynamic ground states, or merely frozen accidents — metastable configurations trapped by rapid cooling? In June 2025, researchers at the University of Michigan used large-scale quantum-mechanical simulations to show that certain quasicrystalline arrangements are genuine enthalpy-stabilized ground states. They are not frozen mistakes. Nature, given unlimited time and thermal equilibrium, would still choose the quasicrystal. The aperiodic arrangement is not a compromise — it is the answer.