Periodic boundary conditions on the pseudosphere

We provide a framework to build periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary conditions in the Euclidean plane, we introduce all the needed mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane. We stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems and we illustrate how to implement such periodic boundary conditions in two examples, the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices.Comment: 30 pages, minor corrections, accepted to J. Phys.

Pentagon-Based Radial Tiling with Triangles and Rectangles and Its Spatial Interpretation

The paper considers a type of radial pentagon-based tiling consisting of two shapes: triangle and rectangle. The ob tained solution has a spatial interpretation in a 3D arrangement of equilateral triangles and squares dictated by the particular array of concave cupolae of the second sort, minor type (CC-II 5.m). These cupolae are arranged so that their decagonal bases partly overlap, making a pentagonal pattern (similar to the one of the Penrose tiling). Covering the folds between the faces of such a polyhedral structure with polygons, we use exactly equi lateral triangles and squares, thanks to the trigonometric prop erties of CC-II-5.m. Observed in the orthogonal projection onto the plane of the polygonal bases, this 3D “covering” is viewed as a pentagonal-based radial tiling in the Euclidean plane. Equilateral triangles will be projected into congruent isosceles triangles corresponding to those obtained by the radial sec tion of a regular pentagon in 5 parts. The squares are project ed into rectangles whose ratio is: a:b = 1:φ/√(1+φ2), where φ is the golden ratio. These triangles and rectangles form a ra dial tiling consisting of 5 sectors of the plane, where the pat terns of the established tiles are repeated locally periodically. However, with 5-fold rotation of the pattern, the tiling itself is non-periodic. The various tiling solutions that can be obtained in this way may serve as inspiration for the geometric design, especially interesting in architecture and applied arts, e.g. for rosettes, brise soleils, mosaics, stained glass, fences, partition screens and the likehttps://smartart-conference.rs/sr/%d1%82%d0%b5%d0%bc%d0%b0%d1%82%d1%81%d0%ba%d0%b8-%d0%b7%d0%b1%d0%be%d1%80%d0%bd%d0%b8%d0%ba-2021/ http://doi.fil.bg.ac.rs/pdf/eb_ser/smartart/2022-2/smartart-2022-2-ch19.pd