The paper introduces conformal tilings, wherein tiles have specified conformal shapes. The principal example involves conformally regular pentagons which tile the plane in a pattern generated by a subdivision rule. Combinatorial symmetries imply rigid conformal symmetries, which in turn illustrate a new type of tiling self-similarity. In parallel with the conformal tilings, the paper develops d...

Spatially homogeneous random tessellations that are stable under iteration (nesting) in the 3-dimensional Euclidean space are considered, so-called STIT tessellations. They arise as outcome of a spatio-temporal process of subsequent cell division and consequently they are not facet-to-facet. The intent of this paper is to develop a detailed analysis of the combinatorial structure of such tessel...

Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids (mass centers) of the corresponding Voronoi regions with respect to a given density function. Centroidal Voronoi tessellations may also be defined in more abstract and more general settings. Due to the natural optimization pr...

We observe stationary random tessellations X = {Ξn}n≥1 in R d through a convex sampling window W that expands unboundedly and we determine the total (k − 1)-volume of those (k − 1)-dimensional manifold processes which are induced on the k-facets of X (1≤ k ≤ d− 1) by their intersections with the (d− 1)-facets of independent and identically distributed motioninvariant tessellations Xn generated ...

In this paper we survey results about affinely regular polygons. First, the definitions and classification of affinely regular polygons are given. Then the theory of Bachmann–Schmidt is outlined. There are several classical theorems about regular polygons, some of them having analogues in finite planes, such as the Napoleon–Barlotti theorem. Such analogues, variants of classical theorems are al...

Abstract In this paper, using both hand-drawn and computer-drawn graphics, we establish a method to generate advanced Escher-like spiral tessellations. We first give way achieve simple tilings of cyclic symmetry. Then, introduce several conformal mappings three derived tilings. To obtain tessellations on the generated tilings, given pre-designed wallpaper motifs, specify tessellations’ implemen...

It is well known that two regular tetrahedra can be combined with a single regular octahedron to tile (complete fill) three-dimensional Euclidean space . This structure was called the "octet truss" by Buckminster Fuller. It was believed that such a tiling, which is the Delaunay tessellation of the face-centered cubic (fcc) lattice, and its closely related stacking variants, are the only tessell...

The principles of origami design have proven useful in a number technological applications. Origami tessellations particular constitute class morphing metamaterials with unusual geometric and elastic properties. Although inextensible principle, fine creases allow to effectively deform non-isometrically. Determining the strains that are compatible coarse-grained kinematics as well corresponding ...