a, Numerical eigenenergy spectra for the heptagon-kagome lattice versus system size. The smallest layout tiling consists of a central heptagon and a shell of seven immediate neighbours (one shell). Each successive tiling includes another shell of immediate neighbours of the previous one. The spectra are plotted for one (red), two (blue), three (cyan) and four (gold) shells of neighbours. b–e, The results for each individual system size. The density of states converges rapidly with number of shells despite relatively small system sizes and effective hard-wall confinement. The theoretical plots elsewhere in this paper are from three-shell simulations. The experimental device consists of two shells.