Distortions are ubiquitous in nature. Under perturbations such as stresses, fields, or other changes, a physical system reconfigures by following a path from one state to another; this path, often a collection of atomic trajectories, describes a dist
ortion. Here we introduce an antisymmetry operation called distortion reversal, 1*, that reverses a distortion pathway. The symmetry of a distortion pathway is then uniquely defined by a distortion group involving 1*; it has the same form as a magnetic group that involves time reversal, 1. Given its isomorphism to magnetic groups, distortion groups could potentially have commensurate impact in the study of distortions as the magnetic groups have had in the study of magnetic structures. Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways, and interface dynamics.
Pulli kolam is a ubiquitous art form in south India. It involves drawing a line looped around a collection of dots (pullis) place on a plane such that three mandatory rules are followed: all line orbits should be closed, all dots are encircled and no
two lines can overlap over a finite length. The mathematical foundation for this art form has attracted attention over the years. In this work, we propose a simple 5-step topological method by which one can systematically draw all possible kolams for any number of dots N arranged in any spatial configuration on a surface.
Rotation-reversal symmetry was recently introduced to generalize the symmetry classification of rigid static rotations in crystals such as tilted octahedra in perovskite structures and tilted tetrahedral in silica structures. This operation has impor
tant implications for crystallographic group theory, namely that new symmetry groups are necessary to properly describe observations of rotation-reversal symmetry in crystals. When both rotation-reversal symmetry and time-reversal symmetry are considered in conjunction with space group symmetry, it is found that there are 17,803 types of symmetry, called double antisymmetry, which a crystal structure can exhibit. These symmetry groups have the potential to advance understanding of polyhedral rotations in crystals, the magnetic structure of crystals, and the coupling thereof. The full listing of the double antisymmetry space groups can be found in the supplemental materials of the present work and online at our website: http://sites.psu.edu/gopalan/research/symmetry/
For over a century, the structure of materials has been described by a combination of rotations, rotation
