Related papers: The Hexagonal Tiling Honeycomb
We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with any irrational angle in degree: they are three $1$-parameter families of pentagonal subdivisions of the Platonic solids, with…
The top of the attractor $A$ of a hyperbolic iterated function system $\left\{ f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}|i=1,2,\dots,M\right\} $ is defined and used to extend self-similar tilings to overlapping systems. The theory…
A convex triangular grid is represented by a planar digraph $G$ embedded in the plane so that (a) each bounded face is surrounded by three edges and forms an equilateral triangle, and (b) the union $\Rscr$ of bounded faces is a convex…
We compute the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the central rhombus and the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the `almost central` rhombus…
We construct a unilateral lattice tiling of $\mathbb{R}^n$ into hypercubes of two differnet side lengths $p$ or $q$. This generalizes the Pythagorean tiling in $\mathbb{R}^2$. We also show that this tiling is unique up to symmetries, which…
In contrast to many known results concerning periodic tilings of the Euclidean plane with pentagons, here tilings with rotational symmetry are investigated. A certain class of convex pentagons is introduced. It can be shown that for any…
The honeycomb problem on the sphere asks for the perimeter-minimizing partition of the sphere into N equal areas. This article solves the problem when N=12. The unique minimizer is a tiling of 12 regular pentagons in the dodecahedral…
Convex hexagons that can tile the plane have been classified into three types. For the generic cases (not necessarily convex) of the three types and two other special cases, we classify tilings of the plane under the assumption that all…
Atkinson [2] found a sequence of three-dimensional hyperbolic polyhedra whose dihedral angles are $\pi /3$. In this paper, we construct another sequence of such polyhedra. We also determine the volumes of some of these polyhedra.
A combinatorial tiling of the sphere is naturally given by an embedded graph. We study the case that each tile has exactly five edges, with the ultimate goal of classifying combinatorial tilings of the sphere by geometrically congruent…
It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more…
Material's geometrical structure is a fundamental part of their properties. The honeycomb geometry of graphene is responsible for the arising of its Dirac cone, while the kagome and Lieb lattice hosts flat bands and pseudospin-1 Dirac…
A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, $n \geq 5$, $n \neq 28$, can be tiled with similar right triangles, then one of the angles of these triangles is in…
This paper views the honeycomb conjecture and the Kepler problem essentially as extreme value problems and solves them by partitioning 2-space and 3-space into building blocks and determining those blocks that have the universal extreme…
We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two…
Since the thesis of K. Reinhardt in 1918, it is well known that there are exactly three types of convex hexagons that can tile the plane. However, the proof of the fact is far from being complete. We prove this fact, under an assumption…
We compute the number of rhombus tilings of a hexagon with sides $N,M,N,N,M,N$, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length $M$.
This presentation starts with the regular polygons, of course, then with the Platonic and Archimedean solids. The latter ones are whose symmetry groups are transitive on the vertices, and in addition, whose faces are regular polygons (see…
We develop the basic tools for classifying edge-to-edge tilings of the sphere by congruent pentagons. Then we prove that, for the edge combination $a^2b^2c$, such tilings are three two-parameter families of pentagonal subdivisions of the…
In a convex mosaic in $\mathbb{R} ^d$ we denote the average number of vertices of a cell by $\bar v$ and the average number of cells meeting at a node by $\bar n$. Except for the $d=2$ planar case, there is no known formula prohibiting…