Fig. 1

Fibonacci square grid, LEED and STM. a The Fibonacci square grid, with highlighted constituent tiles (blue: L × L, red: S × S, and green: S × L). The centre of rotation of four-fold symmetry is marked by a circle. b Low-energy electron diffraction pattern (60 eV, inverted for clarity) from the clean two-fold surface of the icosahedral (i) Al–Pd–Mn quasicrystal. Diffraction spots are τ-scaled in the primary two-fold axes (highlighted with white circles), with two reciprocal space lengths indicated: a = (14.1 ± 0.3) nm⁻¹ and b = (22.5 ± 0.3) nm⁻¹. c Quasicrystalline C₆₀ on the two-fold i-Al–Pd–Mn quasicrystal as imaged by scanning tunnelling microscopy. Scale bar represents 4 nm. d Image c overlaid with a Fibonacci square grid of S = 1.26 nm, L = 2.04 nm (black lines) to highlight the ordering of C₆₀. A τ-deflated Fibonacci square grid of L′ = L/τ = S and S′ = S/τ is shown by white lines on the right-hand side. Note that the original grid overlaps with the τ-deflated grid