Periodic graph (geometry)

About: Periodic graph (geometry) is a research topic. Over the lifetime, 117 publications have been published within this topic receiving 1916 citations.

A computational model for periodic pattern perception based on frieze and wallpaper groups

Abstract: We present a computational model for periodic pattern perception based on the mathematical theory of crystallographic groups. In each N-dimensional Euclidean space, a finite number of symmetry groups can characterize the structures of an infinite variety of periodic patterns. In 2D space, there are seven frieze groups describing monochrome patterns that repeat along one direction and 17 wallpaper groups for patterns that repeat along two linearly independent directions to tile the plane. We develop a set of computer algorithms that "understand" a given periodic pattern by automatically finding its underlying lattice, identifying its symmetry group, and extracting its representative motifs. We also extend this computational model for near-periodic patterns using geometric AIC. Applications of such a computational model include pattern indexing, texture synthesis, image compression, and gait analysis.

Periodic Euclidean Solutions and the Finite Temperature Yang-Mills Gas

Abstract: We present explicit periodic solutions for SU(2) Euclidean gauge theory and briefly consider the contribution of the corresponding finite-temperature configurations to the partition function of the Yang-Mills gas

A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics.

Abstract: Herein we describe some properties and the occurrences of a beautiful geometric figure that is ubiquitous in chemistry and materials science, however, it is not as well-known as it should be. We call attention to the need for mathematicians to pay more attention to the richly structured natural world, and for materials scientists to learn a little more about mathematics. Our account is informal and eschews any pretence of mathematical rigor, but does start with some necessary mathematics. Regular figures such as the five regular Platonic polyhedra are an enduring part of human culture and have been known and celebrated for thousands of years. Herein we consider them as the five regular tilings on the surface of a sphere (a two-dimensional surface of positive curvature). A flag of a tiling of a two-dimensional surface consists of a combination of a coincident tile, edge, and vertex. A generally accepted definition of regularity is flag transitivity, which means that all flags are related by symmetries of the tiling (i.e. there is just one kind of flag). In addition to the five Platonic solids, there are three regular tilings of the plane (a surface of zero curvature), and these are the familiar coverings of the plane by triangles, squares, or hexagons tiled edge-to-edge. The corresponding regular tilings of three-dimensional space are also wellknown. Flags are now a polyhedron (tile) with a coincident face, edge, and vertex, and the regular tilings of the three-sphere are the six nonstellated regular polytopes of four dimensional space. We remark that four dimensions is the richest space in this regard; higher dimensions have only three regular polytopes (and of course three dimensions has five). However, in flat threedimensional (Euclidean) space, the space of our day-to-day experience, there is disappointingly only one regular tiling—the familiar space filling by cubes sharing faces (face-to-face). The classic reference to these figures is Coxeter!s Regular Polytopes, in which he remarks on the tilings of threedimensional Euclidean space: “For the development of a general theory, it is an unhappy accident that only one honeycomb [tiling] is regular...”. Unhappy indeed, because, perhaps as a consequence, the rich world of periodic graphs, which are the underlying topology of crystal structures, has been largely neglected by mathematicians. The graph associated with (carried by) the regular tiling by cubes is the set of edges and vertices. It is notably the structure of a form of elemental polonium, and chemists often refer to it as the a-Po net. Recently a system of symbols for nets has been developed and this net has the symbol pcu. Our review is concerned with another such periodic graph, and an associated surface.

Mining Periodic Behavior in Dynamic Social Networks

Abstract: Social interactions that occur regularly typically correspond to significant yet often infrequent and hard to detect interaction patterns. To identify such regular behavior, we propose a new mining problem of finding periodic or near periodic subgraphs in dynamic social networks. We analyze the computational complexity of the problem, showing that, unlike any of the related subgraph mining problems, it is polynomial. We propose a practical, efficient and scalable algorithm to find such subgraphs that takes imperfect periodicity into account. We demonstrate the applicability of our approach on several real-world networks and extract meaningful and interesting periodic interaction patterns.