The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form tile-2-transitive or tile-3-transitive tilings. In this article, we present some combinatorial results concerning pentagons that admit $i$-block transitive tilings for $i \in \mathbb{N}$. These results form the basis for an automated approach to finding all pentagons that admit $i$-block transitive tilings for each $i \in \mathbb{N}$. We will present the methods of this algorithm and the results of the computer searches so far, which includes a complete classification of all pentagons admitting 1-, 2-, and 3-block transitive tilings, among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling.
SummaryWe present a generalization of the Voderberg tile, which, in addition to admitting periodic and nonperiodic spiral tilings of the plane, has the property that just two copies can surround 1 or 2 copies of the tile. We construct a generalization of this tile that admits periodic and nonperiodic spiral tilings of the plane while enjoying the property that any number of copies of the tile can be surrounded by just 2 copies. In doing so, we solve two open problems posed in the classic book Tilings and Patterns by Grünbaum and Shephard.
In this article, the results of a computerized search for edge-marked polyforms (tiles formed from either equilateral triangles, squares, or regular hexagons) with finite Heesch number are presented.
This work is motivated by a paper of Huh and Oh, in which the authors prove that the minimum number of sticks required to form a knot in ℤ 3 is 12. In this article the authors prove that the stick number in the simple hexagonal lattice is 11. Moreover, the stick number of the trefoil in the simple hexagonal lattice is 11.
D. Schattschneider proved that there are exactly eight unilateral and equitransitive tilings of the plane by squares of three distinct sizes. This article extends Schattschneider’s methods to determine a classification of all such tilings by squares of four different sizes. It is determined that there are exactly 39 unilateral and equitransitive tilings by squares of four different sizes.
We show that $ \{0, \varphi+3, -3\varphi+2, -\varphi+\frac{5}{2}\} $ is the set of slopes of nonexpansive directions for a minimal subshift in the Jeandel-Rao Wang shift, where $ \varphi = (1+\sqrt{5})/2 $ is the golden mean. This set is a topological invariant allowing to distinguish the Jeandel-Rao Wang shift from other subshifts. Moreover, we describe the combinatorial structure of the two resolutions of the Conway worms along the nonexpansive directions in terms of irrational rotations of the unit interval. The introduction finishes with pictures of nonperiodic Wang tilings corresponding to what Conway called the cartwheel tiling in the context of Penrose tilings. The article concludes with open questions regarding the description of octopods and essential holes in the Jeandel-Rao Wang shift.
Convex polygons, shapes bounded by straight lines in which all of the corners point outward, are the simplest of shapes, but despite their simplicity, there are many unsolved problems concerning polygons. Some polygons have the nice property that they fit together espcially well, so that you can use lots of copies of them to cover a large surface to form what is called a tiling. For some polygons, such as triangles, it is easy to see how to form tilings. For others, such as seven-sided convex polygons, it is impossible to form a tiling. This article discusses the history of a long-standing open question in geometry: which convex pentagons give rise to tilings of the plane? The authors also discuss their contribution to the solution of this problem, which involved developing a computerized algorithm to help them search for a new kind of convex pentagon that can form tilings, and the basic idea of this algorithm is discussed.
