Shapes of polyhedra and triangulations of the sphere

The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic cone-manifold. In some interesting special cases, the metric completion is an orbifold. The concrete description of these spaces of shapes gives information about the combinatorial classification of triangulations of the sphere with no more than 6 triangles at a vertex.Comment: 39 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper25.abs.htm

On Fenchel-Nielsen coordinates on Teichm\"uller spaces of surfaces of infinite type

We introduce Fenchel-Nielsen coordinates on Teicm\"uller spaces of surfaces of infinite type. The definition is relative to a given pair of pants decomposition of the surface. We start by establishing conditions under which any pair of pants decomposition on a hyperbolic surface of infinite type can be turned into a geometric decomposition, that is, a decomposition into hyperbolic pairs of pants. This is expressed in terms of a condition we introduce and which we call Nielsen convexity. This condition is related to Nielsen cores of Fuchsian groups. We use this to define the Fenchel-Nielsen Teichm\"uller space associated to a geometric pair of pants decomposition. We study a metric on such a Teichm\"uller space, and we compare it to the quasiconformal Teichm\"uller space, equipped with the Teichm\"uller metric. We study conditions under which there is an equality between these Teichm\"uller spaces and we study topological and metric properties of the identity map when this map exists

Shape and symmetry determine two-dimensional melting transitions of hard regular polygons

The melting transition of two-dimensional (2D) systems is a fundamental problem in condensed matter and statistical physics that has advanced significantly through the application of computational resources and algorithms. 2D systems present the opportunity for novel phases and phase transition scenarios not observed in 3D systems, but these phases depend sensitively on the system and thus predicting how any given 2D system will behave remains a challenge. Here we report a comprehensive simulation study of the phase behavior near the melting transition of all hard regular polygons with $3\leq n\leq 14$ vertices using massively parallel Monte Carlo simulations of up to one million particles. By investigating this family of shapes, we show that the melting transition depends upon both particle shape and symmetry considerations, which together can predict which of three different melting scenarios will occur for a given $n$. We show that systems of polygons with as few as seven edges behave like hard disks; they melt continuously from a solid to a hexatic fluid and then undergo a first-order transition from the hexatic phase to the fluid phase. We show that this behavior, which holds for all $7\leq n\leq 14$, arises from weak entropic forces among the particles. Strong directional entropic forces align polygons with fewer than seven edges and impose local order in the fluid. These forces can enhance or suppress the discontinuous character of the transition depending on whether the local order in the fluid is compatible with the local order in the solid. As a result, systems of triangles, squares, and hexagons exhibit a KTHNY-type continuous transition between fluid and hexatic, tetratic, and hexatic phases, respectively, and a continuous transition from the appropriate "x"-atic to the solid. [abstract truncated due to arxiv length limitations]