Visualizing the topological pentagon states of a giant C₅₄₀ metamaterial

Systems with broken continuous symmetry in ideal lattices cannot be rectified through rearrangement or deformation. Topological metamaterials featuring nontrivial, artificially induced phase transitions have emerged as pivotal constituents for engineering these topological defects, which, until now, have mostly been experimentally realized in linear or planar configurations. Buckminster Fuller lent his name to the C₆₀ ball-shaped carbon allotrope, which is not only the roundest molecule in existence but also embodies 3D topological defects. Here, we construct a C₅₄₀ metamaterial composed of interspersed pentagons in a hexagonal network of hollow tubes and cavities. By 3D printing this giant closed-cage topology, the nontrivial state-confinements can be fully controlled and visualized, which, in contrast, in synthesized or naturally found fullerenes, is highly challenging. Thanks to our macroscopic metamaterials approach, we are able to map in real-space topological pentagon states probed by sound waves. Our results show how a seemingly unrelated approach can unveil deep physical understanding in carbon allotropes and potentially in a plethora of other complex systems in the near future.