Fig. 2: Ground-state degeneracy of the triangular ball-stick system.

a Two isotopic structures of the unit-cell, splitting from the original space-filling scheme for the hard triangle. b Crystalline morphology. c Spatial arrangement of unit-cells. One of \({\alpha }_{1}\) and \({\alpha }_{2}\), which are the angles formed by two unit-cell primitive vectors, is larger than \(\pi /2\), corresponding to a rhombic crystal structure. Each point represents a unit-cell composed of two monomers with opposite body-orientations, as shown at the top of this picture. d Three stable local structures composed of the two isotopes with the local clusters in the three configurations all composed of six atoms, which are circled by red, purple, and green frames, respectively. e Three typical small hand-made structures constructed based on the three local structures in d. The top one is a part of crystal, noted as ‘crystal’, the middle one is a mixture of the first two local configurations in d, noted as ‘perfect-mixing’, and the bottom one is formed by mixing all the three local configurations in d, noted as ‘tower-like’. The local clusters are framed with the same color as in d. f Potential energies (\({E}_{{{{\rm{p}}}}}\)) of different structures under various cutoff radii (\({r}_{{{{\rm{c}}}}}\)) with the solid lines representing \({E}_{{{{\rm{p}}}},{{{\rm{lowest}}}}}\) and the dashed lines representing \({E}_{{{{\rm{p}}}},{{{\rm{avg}}}}}\). Physical quantities are expressed in L-J units.