We study lattice trees, lattice animals, and percolation on non-Euclidean lattices that correspond to regular tessellations of twoand three-dimensional hyperbolic space. We prove that critical exponents of these models take on their mean field values. Our methods are mainly combinatorial and geometric.

A decoration of a hyperbolic surface finite type is choice circle, horocycle or hypercycle about each cone-point, cusp flare the surface, respectively. In this article we show that induces unique canonical tessellation and dual decomposition underlying surface. They are analogues weighted Delaunay Voronoi in Euclidean plane. We develop characterisation terms geometric equivalents Delaunay's emp...

Euclidean tilings are constantly applied to many fields of engineering (mechanical, civil, chemical, etc.). These tessellations usually named after Cundy & Rollett’s notation. However, this notation has two main problems related ambiguous conformation and uniqueness. This communication explains the GomJau-Hogg’s for generating all regular, semi-regular (uniform) demi-regular (k-uniform, up ...

PRÉCIS There are various structural morphologies. Some of these may usefully be regarded as ““natural”” morphologies of structure. Characteristically they are discovered and elaborated in accord with their intrinsic quality, rather than idealistically schematized and then imposed upon existence. Examples are the regular and semi-regular polyhedra, the beautiful pattern of interrelationship they...

In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the wellknown classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a ...

We present the plane-sweep incremental algorithm, a hybrid approach for computing Delaunay tessellations of large point sets whose size exceeds the computer’s main memory. This approach unites the simplicity of the incremental algorithms with the comparatively low memory requirements of plane-sweep approaches. The procedure is to first sort the point set along the first principal component and ...

Abstract Origami tessellations are geometric pieces folded from a single sheet of paper with flatly overlapped facets. Most existing origami constructed by first marking grid crease lines on the and then arranging repeating patterns along grid. However, this design method is limited because it cannot that be represented grid, such as regular pentagon. This proposes new construction for solves p...

About 300 years ago Sébastien Truchet systematically studied patterns that could be formed from square tiles that were divided by a diagonal into a white triangle and a black triangle. Other pattern creators have been inspired by him to make Truchet-like tilings composed of circular arcs and other motifs. These tilings are all based on Euclidean tessellations, usually the tiling by squares. In ...

Many traditional and new musical instruments make use of an isomorphic note layout across a uniform planar tessellation. Recently, a number of hexagonal isomorphic keyboards have become available commercially. Each such keyboard or interface uses a single specific layout for notes, with specific justifications as to why this or that layout is better. This paper is an exploration of all possible...

In this work, we consider the tessellations (or tilings) of Euclidean and hyperbolic planes using copies of a regular polygon. We introduce the concept of k-type of vertices and edges, which allow a thorough control of these elements when the tessellation increases, and we obtain an enumeration for the vertices, edges, and polygons at a given distance. Partially funded by a grant from Fapesp (2...