Bio-inspired programmable metamaterials with logarithmic spiral hierarchy for energy absorption

• A rotation-governed self-similar chiral hierarchical honeycomb is constructed via an explicit rotate–tangential-extend–closure geometric rule, enabling recursive multilevel nesting. • The hierarchical evolution of cell vertices is quantitatively described by logarithmic spirals r = a · e b θ , linking the rotation angle α to a tunable radial scale gradient. • The spiral parameters q α and b α reveal how geometric chirality and hierarchy jointly govern multistage buckling and densification dela