Tessellations Of Surfaces Research Articles

Decorated Discrete Conformal Maps and Convex Polyhedral Cusps

Abstract We discuss a notion of discrete conformal equivalence for decorated piecewise Euclidean surfaces (PE-surface), that is, PE-surfaces with a choice of circle about each vertex. It is closely related to inversive distance and hyperideal circle patterns. Under the assumption that the circles are non-intersecting, we prove the corresponding discrete uniformization theorem. The uniformization theorem for discrete conformal maps corresponds to the special case that all circles degenerate to points. Our proof relies on an intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces and convex hyperbolic polyhedra. It is based on a concave variational principle, which also provides a method for the computation of decorated discrete conformal maps.

Solvent and relative humidity effect on highly ordered polystyrene honeycomb patterns analyzed by Voronoi tesselation

ABSTRACTHighly ordered honeycomb‐patterned polystyrene surfaces are efficiently prepared by static breath figure method. The structured arrays can be obtained by casting a dilute solution polymer on glass substrates under various conditions. Tetrahydrofuran and chloroform are used as solvent to form cavities of several micrometers. The analysis of the surfaces indicates nonlinear relation between concentration and pore size in this system. Voronoi tessellations of the polystyrene surfaces at different relative humidity (RH) are achieved, and each conformational entropy determined. Optimum parameters of concentration and RH are obtained for both solvents. Analysis of hole size distribution and conformational entropy demonstrates the high order of the films obtained. This is a promising method for the fabrication of homogeneous and highly porous films from polystyrene. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016, 133, 44004.

Gauge theories, tessellations & Riemann surfaces

We study and classify regular and semi-regular tessellations of Riemann surfaces of various genera and investigate their corresponding supersymmetric gauge theories. These tessellations are generalizations of brane tilings, or bipartite graphs on the torus as well as the Platonic and Archimedean solids on the sphere. On higher genus they give rise to intricate patterns. Special attention will be paid to the master space and the moduli space of vacua of the gauge theory and to how their geometry is determined by the tessellations.

Classical simulation of measurement-based quantum computation on higher-genus surface-code states

We consider the efficiency of classically simulating measurement-based quantum computation on surface-code states. We devise a method for calculating the elements of the probability distribution for the classical output of the quantum computation. The operational cost of this method is polynomial in the size of the surface-code state, but in the worst case scales as ${2}^{2g}$ in the genus $g$ of the surface embedding the code. However, there are states in the code space for which the simulation becomes efficient. In general, the simulation cost is exponential in the entanglement contained in a certain effective state, capturing the encoded state, the encoding, and the local postmeasurement states. The same efficiencies hold, with additional assumptions on the temporal order of measurements and on the tessellations of the code surfaces, for the harder task of sampling from the distribution of the computational output.

Surface Quotients of Hyperbolic Buildings

Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph K(v,v). We investigate and mostly determine the set of triples (p,v,g) for which there exists a uniform lattice {\Gamma} in Aut(I(p,v)) such that {\Gamma}\I(p,v) is a compact orientable surface of genus g. Surprisingly, the existence of {\Gamma} depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction of {\Gamma}, together with a theorem of Haglund, implies that for p>=6, every uniform lattice in Aut(I) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.

Platonic tessellations of Riemann surfaces as models in chemistry: non-zero genus analogues of regular polyhedra

The concept of regular polyhedra can be generalized to regular polygonal networks on surfaces of non-zero genus, i.e. platonic tessellations of Riemann surfaces. Examples of such figures include the genus 3 Klein {7,3}8 tessellation of 24 heptagons and the Dyck {8,3}6 tessellation of 12 octagons used to describe possible allotropes of carbon and boron nitride exhibiting low density negative curvature zeolite-like structures. The dual of a related genus 3 double cube {4,2·3} tessellation consisting of four hexagons meeting at each vertex is used by Sadoc and Charvolin in their topological theory of bilayers in liquid crystal and micellar structures. Similar platonic tessellations of non-zero genus provide geometric models for the double point groups used in molecular physics such as the genus 2 {3,4+4} tessellation for the double octahedral group and the genus 5 {3,2·5} tessellation for the double icosahedral group.

Constrained Centroidal Voronoi Tessellations for Surfaces

Centroidal Voronoi tessellations are useful for subdividing a region in Euclidean space into Voronoi regions whose generators are also the centers of mass, with respect to a prescribed density function, of the regions. Their extensions to general spaces and sets are also available; for example, tessellations of surfaces in a Euclidean space may be considered. In this paper, a precise definition of such constrained centroidal Voronoi tessellations (CCVTs) is given and a number of their properties are derived, including their characterization as minimizers of an "energy." Deterministic and probabilistic algorithms for the construction of CCVTs are presented and some analytical results for one of the algorithms are given. Computational examples are provided which serve to illustrate the high quality of CCVT point sets. Finally, CCVT point sets are applied to polynomial interpolation and numerical integration on the sphere.

Torsion free subgroups of fuchsian groups and tessellations of surfaces

Torsion free subgroups of fuchsian groups and tessellations of surfaces

Regular Tessellations of Surfaces and (p, q, 2)-Triangle Groups

The purpose of this paper is a study of the existence and classification of certain regular tessellations of surfaces and their relationship with the Schwarz triangle groups of type (p, q, 2). These tessellations are the natural generalizations to arbitrary surfaces of the spherical tessellations corresponding to the five Platonic Solids. A regular tessellation of type {p, q}, a {p, q}-pattern, for short, on a surface M is a tessellation of M by p-sided faces such that the valence of each vertex is q. (A more precise definition appears in ? 1.) No global symmetry is assumed. A { p, q}-pattern on M is said to be geometric if M admits a Riemannian metric of constant curvature with respect to which the edges of the pattern are geodesic arcs of equal length and the interior angles at all vertices in all faces are (2 7/q). If the curvature happens to he zero then we further insist that the area of each face be 1. It is natural to call two {p, q}-patterns on M

Torsion free subgroups of Fuchsian groups and tessellations of surfaces

Torsion free subgroups of Fuchsian groups and tessellations of surfaces