Ground states of a collection of N point-like particles constrained to lie on the surface of a torus and interacting via a cubic potential. The lattices are labeled by (r,N), where r is the ratio between the largest and smallest radius of the torus. The figure shows prominent example of topological defects such as dislcinations (5-fold disclinations are marked in red and 7-fold disclinations in blue), dislocations and grain-boundaries. While in planar crystals these defects are energetically costly and don't appear in the ground state, on a surface of non-zero Gaussian curvature defects proliferate in order to balance the elastic strain introduced by the curvature of the underlying medium. L. Giomi and M. J. Bowick, Eur. Phys. J. E 27, 275 (2008) & M. Bowick and L. Giomi, Adv. Phys. 58, 449 (2009).

Image courtesy: Dr. L. Giomi, School of Engineering and Applied Science, Harvard University.