Quasicrystals (Shechtman 1984, Nobel 2011) produce sharp diffraction peaks with 5-fold / 10-fold symmetry forbidden by classical crystallography. The pattern arises from a cut of a periodic lattice in higher dimensions.
Quasicrystal Real-Space Structure
Diffraction Pattern |F(q)|²
Key insight: Peaks at positions q = Σ nᵢ b⃗ᵢ where bᵢ are incommensurate basis vectors (5 for Penrose). Sharp peaks ↔ long-range positional order. 5-fold symmetry ↔ forbidden crystal. Dense peak set ↔ never truly periodic — but indexable with 4-5 Miller indices.