An integer representation for periodic tilings of the plane by regular polygons

We describe a representation for periodic tilings of the plane by regular polygons. Our approach is to represent explicitly a small subset of seed vertices from which we systematically generate all elements of the tiling by translations. We represent a tiling concretely by a (2+n)×4 integer matrix containing lattice coordinates for two translation vectors and n seed vertices. We discuss several properties of this representation and describe how to exploit the representation elegantly and efficiently for reconstruction, rendering, and automatic crystallographic classification by symmetry detection

The application of the principles of symmetry to the synthesis of multi-coloured counterchange patterns

Attention is focused on the theoretical principles governing the underlying geometry of motifs, border patterns and all-over patterns. The systematic classification and construction of two-dimensional periodic patterns and tilings is introduced, with particular relerence to two-colour and higher colour counterchange possibilities. An identification is made of the geometrical restraints encountered when introducing systematic interchange of colour. A wide ranging series of original patterns and tilings is constructed and fully illustrated; these designs have been printed in fabric form and are presented in the accompanying exhibition

Gauge Theories, Toric Varieties and Machine Learning

We discuss some recent study on quiver gauge theories in the setting of toric geometry. After mentioning some basic geometric and topological properties, we consider some mathematical concepts, namely the Mahler measure and the dessins d’enfants, in this context. We then focus on the quiver BPS algebras and their connections to diferent aspects in physics. We also have a discussion on the stability of chiral rings for more general geometry. Besides, we make some comments on the applications of machine learning to relevant topics. This thesis is based on the works [1–16]

Modelado jerárquico de objetos 3D con superficies de subdivisión

Las SSs (Superficies de Subdivisión) son un potente paradigma de modelado de objetos 3D (tridimensionales) que establece un puente entre los dos enfoques tradicionales a la aproximación de superficies, basados en mallas poligonales y de parches alabeados, que conllevan problemas uno y otro. Los esquemas de subdivisión permiten definir una superficie suave (a tramos), como las más frecuentes en la práctica, como el límite de un proceso recursivo de refinamiento de una malla de control burda, que puede ser descrita muy compactamente. Además, la recursividad inherente a las SSs establece naturalmente una relación de anidamiento piramidal entre las mallas / NDs (Niveles de Detalle) generadas/os sucesivamente, por lo que las SSs se prestan extraordinariamente al AMRO (Análisis Multiresolución mediante Ondículas) de superficies, que tiene aplicaciones prácticas inmediatas e interesantísimas, como la codificación y la edición jerárquicas de modelos 3D. Empezamos describiendo los vínculos entre las tres áreas que han servido de base a nuestro trabajo (SSs, extracción automática de NDs y AMRO) para explicar como encajan estas tres piezas del puzzle del modelado jerárquico de objetos de 3D con SSs. El AMRO consiste en descomponer una función en una versión burda suya y un conjunto de refinamientos aditivos anidados jerárquicamente llamados "coeficientes ondiculares". La teoría clásica de ondículas estudia las señales clásicas nD: las definidas sobre dominios paramétricos homeomorfos a R" o (0,1)n como el audio (n=1), las imágenes (n=2) o el vídeo (n=3). En topologías menos triviales, como las variedades 2D) (superficies en el espacio 3D), el AMRO no es tan obvio, pero sigue siendo posible si se enfoca desde la perspectiva de las SSs. Basta con partir de una malla burda que aproxime a un bajo ND la superficie considerada, subdividirla recursivamente y, al hacerlo, ir añadiendo los coeficientes ondiculares, que son los detalles 3D necesarios para obtener aproximaciones más y más finas a la superficie original. Pasamos después a las aplicaciones prácticas que constituyen nuestros principal desarrollo original y, en particular, presentamos una técnica de codificación jerárquica de modelos 3D basada en SSs, que actúa sobre los detalles 3D mencionados: los expresa en un referencial normal loscal; los organiza según una estructura jerárquica basada en facetas; los cuantifica dedicando menos bits a sus componentes tangenciales, menos energéticas, y los "escalariza"; y los codifica dinalmente gracias a una técnica similar al SPIHT (Set Partitioning In Hierarchical Tress) de Said y Pearlman. El resultado es un código completamente embebido y al menos dos veces más compacto, para superficies mayormente suaves, que los obtenidos con técnicas de codificación progresiva de mallas 3D publicadas previamente, en las que además los NDs no están anidados piramidalmente. Finalmente, describimos varios métodos auxiliares que hemos desarrollado, mejorando técnicas previas y creando otras propias, ya que una solución completa al modelado de objetos 3D con SSs requiere resolver otros dos problemas. El primero es la extracción de una malla base (triangular, en nuestro caso) de la superficie original, habitualmente dada por una malla triangular fina con conectividad arbitraria. El segundo es la generación de un remallado recursivo con conectividad de subdivisión de la malla original/objetivo mediante un refinamiento recursivo de la malla base, calculando así los detalles 3D necesarios para corregir las posiciones predichas por la subdivisión para nuevos vértices

Duality and dynamics of supersymmetric field theories from D-branes on singularities

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2005.Includes bibliographical references (p. 359-373).We carry out various investigations regarding gauge theories on the worldvolume of D-branes probing toric singularities. We first study the connection that arises in Toric Duality between different dual gauge theory phases and the multiplicity of fields in the gauged linear sigma models associated with the probed geometries. We introduce a straightforward procedure for the determination of toric dual theories and partial resolutions based on the (p, q) web description of toric singularities. We study the non-conformal theories that arise in the presence of fractional branes. We introduce a systematic procedure to study the resulting cascading RG flows, including the effect of anomalous dimensions on beta functions. Supergravity solutions dual to logarithmic RG flows are constructed, validating the field theory analysis of the cascades. We systematically study the IR dynamics of cascading gauge theories. We show how the deformation in the dual geometries is encoded in a quantum modification of the moduli space. We construct an infinite family of superconformal quiver gauge theories which are AdS/CFT dual to Sasaki-Einstein horizons with explicit metrics. The gauge theory and geometric computations of R-charges and central charges are shown to agree. We introduce new Type IIB brane constructions denoted brane tilings which are dual to D3-branes probing arbitrary toric singularities. Brane tilings encode both the quiver and superpotential of the gauge theories on the D-brane probes. They give a connection with the statistical model of dimers.(cont.) They provide the simplest known method for computing toric moduli spaces of gauge theories, which reduces to finding the determinant of the Kasteleyn matrix of a bipartite graph.by Sebastián Federico Franco.Ph.D

Mathematical surfaces models between art and reality

In this paper, I want to document the history of the mathematical surfaces models used for the didactics of pure and applied “High Mathematics” and as art pieces. These models were built between the second half of nineteenth century and the 1930s. I want here also to underline several important links that put in correspondence conception and construction of models with scholars, cultural institutes, specific views of research and didactical studies in mathematical sciences and with the world of the figurative arts furthermore. At the same time the singular beauty of form and colour which the models possessed, aroused the admiration of those entirely ignorant of their mathematical attraction

Synthesis and evaluation of geometric textures

Two-dimensional geometric textures are the geometric analogues of raster (pixel-based) textures and consist of planar distributions of discrete shapes with an inherent structure. These textures have many potential applications in art, computer graphics, and cartography. Synthesizing large textures by hand is generally a tedious task. In raster-based synthesis, many algorithms have been developed to limit the amount of manual effort required. These algorithms take in a small example as a reference and produce larger similar textures using a wide range of approaches. Recently, an increasing number of example-based geometric synthesis algorithms have been proposed. I refer to them in this dissertation as Geometric Texture Synthesis (GTS) algorithms. Analogous to their raster-based counterparts, GTS algorithms synthesize arrangements that ought to be judged by human viewers as “similar” to the example inputs. However, an absence of conventional evaluation procedures in current attempts demands an inquiry into the visual significance of synthesized results. In this dissertation, I present an investigation into GTS and report on my findings from three projects. I start by offering initial steps towards grounding texture synthesis techniques more firmly with our understanding of visual perception through two psychophysical studies. My observations throughout these studies result in important visual cues used by people when generating and/or comparing similarity of geometric arrangements as well a set of strategies adopted by participants when generating arrangements. Based on one of the generation strategies devised in these studies I develop a new geometric synthesis algorithm that uses a tile-based approach to generate arrangements. Textures synthesized by this algorithm are comparable to the state of the art in GTS and provide an additional reference in subsequent evaluations. To conduct effective evaluations of GTS, I start by collecting a set of representative examples, use them to acquire arrangements from multiple sources, and then gather them into a dataset that acts as a standard for the GTS research community. I then utilize this dataset in a second set of psychophysical studies that define an effective methodology for comparing current and future geometric synthesis algorithms