Particle Fractionalization and Pleats!
Point defects and Topological defects on Wigner Crystals on Curved Surfaces

Department of Physics 

New York University

Conventionally disclinations relieve curvature stress when a lattice is melded to a curved surface. But we have discovered that “Pleats”, grain boundaries which vanish on the surface (and play a similar role to fabric pleats) can provide a finer control of curvature. Hexagons perfectly tile a flat plane but a soccer ball requires exactly 12 pentagons dispersed among
the hexagons on its curved surface. Pentagons and hexagons are positive and negative topological charges, disclinations, sources for positive and negative curvature. We experimentally investigate the generation of topological charge as flat surfaces are curved. For positive curvature, domes and barrels, there is one pentagon added for every 1/12 of a sphere.
Negative curvature is different! For capillary bridges forming catenoids, unduloids and nodoids pleats relieve the stress before heptagons appear on the surface.