Crystals can only have 2,3,4,6-fold symmetry (crystallographic restriction). Quasicrystals show 5-fold (icosahedral) — "impossible" by classical theory.
Dan Shechtman observed sharp diffraction peaks with icosahedral symmetry in Al-Mn alloy. Nobel Prize 2011. He was ridiculed for years before vindication.
Penrose tiling uses two rhombus shapes in ratio φ = (1+√5)/2 ≈ 1.618. Quasiperiodicity means ordered but non-repeating — like Fibonacci sequences in 2D.