Happy endings for flip graphs

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio

A Skyrme lattice with hexagonal symmetry

Recently it has been found that the structure of Skyrmions has a close analogy to that of fullerene shells in carbon chemistry. In this letter we show that this analogy continues further, by presenting a Skyrme field that describes a lattice of Skyrmions with hexagonal symmetry. This configuration, a novel `domain wall' in the Skyrme model, has low energy per baryon (about 6% above the Faddeev-Bogomolny bound) and in many ways is analogous to graphite. By comparison to the energy per baryon of other known Skyrmions and also the Skyrme crystal, we discuss the possibility of finding Skyrmion shells of higher charge.Comment: 12 pages, 1 figure. To appear in Phys. Lett.

Cones, pringles, and grain boundary landscapes in graphene topology

A polycrystalline graphene consists of perfect domains tilted at angle {\alpha} to each other and separated by the grain boundaries (GB). These nearly one-dimensional regions consist in turn of elementary topological defects, 5-pentagons and 7-heptagons, often paired up into 5-7 dislocations. Energy G({\alpha}) of GB computed for all range 0<={\alpha}<=Pi/3, shows a slightly asymmetric behavior, reaching ~5 eV/nm in the middle, where the 5's and 7's qualitatively reorganize in transition from nearly armchair to zigzag interfaces. Analysis shows that 2-dimensional nature permits the off-plane relaxation, unavailable in 3-dimensional materials, qualitatively reducing the energy of defects on one hand while forming stable 3D-landsapes on the other. Interestingly, while the GB display small off-plane elevation, the random distributions of 5's and 7's create roughness which scales inversely with defect concentration, h ~ n^(-1/2)Comment: 9 pages, 4 figure

Spherical Tiling by 12 Congruent Pentagons

The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-dimensional sphere by 12 congruent pentagons. We find one major class allowing two independent continuous parameters and four classes of isolated examples. The classification is done by first separately classifying the combinatorial, edge length, and angle aspects, and then combining the respective classifications together.Comment: 53 pages, 40 figures, spherical geometr

Grain boundaries in graphene grown by chemical vapor deposition

The scientific literature on grain boundaries (GBs) in graphene was reviewed. The review focuses mainly on the experimental findings on graphene grown by chemical vapor deposition (CVD) under a very wide range of experimental conditions (temperature, pressure hydrogen/hydrocarbon ratio, gas flow velocity and substrates). Differences were found in the GBs depending on the origin of graphene: in micro-mechanically cleaved graphene (produced using graphite originating from high-temperature, high-pressure synthesis), rows of non-hexagonal rings separating two perfect graphene crystallites are found more frequently, while in graphene produced by CVD—despite the very wide range of growth conditions used in different laboratories—GBs with more pronounced disorder are more frequent. In connection with the observed disorder, the stability of two-dimensional amorphous carbon is discussed and the growth conditions that may impact on the structure of the GBs are reviewed. The most frequently used methods for the atomic scale characterization of the GB structures, their possibilities and limitations and the alterations of the GBs in CVD graphene during the investigation (e.g. under e-beam irradiation) are discussed. The effects of GB disorder on electric and thermal transport are reviewed and the relatively scarce data available on the chemical properties of the GBs are summarized. GBs are complex enough nanoobjects so that it may be unlikely that two experimentally produced GBs of several microns in length could be completely identical in all of their atomic scale details. Despite this, certain generalized conclusions may be formulated, which may be helpful for experimentalists in interpreting the results and in planning new experiments, leading to a more systematic picture of GBs in CVD graphene