2D Cross-Section: Surface Atom Fraction in Nanoparticles

This is the 2D-native counterpart to the 3D shell-by-shell nanoparticle model: rather than projecting a 3D Fibonacci-sphere cluster onto the screen, it builds a real 2D triangular lattice — the same atomic arrangement seen in a close-packed crystal plane — ring by ring, from a bare central atom up to a 217-atom disk, using the exact "centered hexagonal number" sequence (6k atoms per ring) that describes real 2D close-packed clusters. As the ring count K increases, live readouts track the total atom count, the physical extent for four real metals (Au, Ag, Pt, Cu) using each one's nearest-neighbour spacing, and what fraction of atoms sit exposed on the rim versus buried in the interior, plus the resulting drop in average coordination number. Because area scales as K² but the rim only as K, this single geometric fact explains — one dimension down from the 3D case — why nanoparticles are dramatically more reactive, more catalytically active, and melt at a lower temperature than the same material in bulk.