Let P be a convex polytope in the spherical space S, in the Euclidean space E, or in the hyperbolic space H. Consider the group GP generated by reflections in the facets of P . We call GP a reflection group generated by P . The problem we consider in this paper is to list polytopes generating discrete reflection groups. The answer is known only for some combinatorial types of polytopes. Already...

High resolution atomic force microscopy (AFM) is used to resolve the evolution of crystallites of a metal organic framework (HKUST-1) grown on Au(111) using a liquid-phase layer-by-layer methodology. The nucleation and faceting of individual crystallites is followed by repeatedly imaging the same submicron region after each cycle of growth and we find that the growing surface is terminated by {...

Bluish-green semipolar GaInN/GaN light emitting diodes (LEDs) were investigated as possible candidates for high-brightness devices even in the long wavelength visible regime. To combine the high material quality known from c-GaN and the advantages of a reduced piezoelectric field, the LED structures were realized on the {11̄01} side facets of selectively grown GaN stripes with triangular cross s...

The microscopic kinetics of ubiquitous solid-solid phase transitions remain poorly understood. Here, by using single-particle-resolution video microscopy of colloidal films of diameter-tunable microspheres, we show that transitions between square and triangular lattices occur via a two-step diffusive nucleation pathway involving liquid nuclei. The nucleation pathway is favoured over the direct ...

Twisted ribbons under tension exhibit a remarkably rich morphology, from smooth and wrinkled helicoids, to cylindrical or faceted patterns. This complexity emanates from the instability of the natural, helicoidal symmetry of the system, which generates both longitudinal and transverse stresses, thereby leading to buckling of the ribbon. Here, we focus on the tessellation patterns made of triang...

This paper addresses the problem of efficiently turning the connectivity of an initial triangular mesh into a new one, using combinatorial edge flip operations only, excluding vertex insertion and deletion. More precisely, we provide a practical but provably good solution for determining a sequence of edge flips between two surface triangulations sharing the same topological genus and the same ...

We report here the epitaxial growth of InGaN/GaN quantum wells on freestanding GaN gratings by molecular beam epitaxy (MBE). Various GaN gratings are defined by electron beam lithography and realized on GaN-on-silicon substrate by fast atom beam etching. Silicon substrate beneath GaN grating region is removed from the backside to form freestanding GaN gratings, and the patterned growth is subse...

A surface is often approximated by a network of triangular facets. In the absence of a precise mathematical description of the underlying surface all information about surface properties such as smoothness and curvature must be inferred from the triangulation itself. Enrichment and coarsening of the surface geometry, for example, can only be carried out if singular features, where C continity i...

A model of the shape of Toutatis based on “high-resolution” radar images obtained in 1992 and 1996 consists of 39,996 triangular facets of roughly equal area, defined by the locations of 20,000 vertices. These define the average spatial resolution of the model as approximately 34 m, significantly finer than Hudson and Ostro’s (1995, Science 270, 84–86) model (1600 vertices, resolution 84 m). Th...

We report the investigation of coherent GaN nanocrystals spontaneously formed by nitrogen-plasma-assisted reactive epitaxy with Ga droplets supported on the single crystalline Si3N4s0001d /Sis111d surface. The distribution of grown GaN nanocrystals, as revealed by scanning electron microscopy and cross-sectional transmission electron microscopy, is very uniform in size s,16 nmd and shape and th...