Convex pentagons that admit i-block transitive tilings

Abstract

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 2block 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 ∈ N. These results form the basis for an automated approach to finding all pentagons that admit i-block transitive tilings for each i ∈ 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.A plane tiling T is a countable family of closed topological disks T = {T 1 , T 2 , ...} that cover the Euclidean plane E 2 without gaps or overlaps; that is, T satisfies 1. i∈N T i = E 2 , and

Can You Pave the Plane with Identical Tiles?

“…In fact, as Reinhardt and several other authors pointed out (see [4,10,12,14]), this theorem can be easily deduced by Euler's formula…”

“…Usually, they are known as Archimedean tilings. In 1619, Kepler enumerated all such tilings as (3,3,3,3,3,3), (3,3,3,3,6), (3,3,3,4,4), (3,3,4,3,4), (3,4,6,4), (3,6,3,6), (3,12,12), (4,4,4,4), (4,6,12), (4,8,8), and (6,6,6). Beautiful illustrations of the Archimedean tilings can be found in many references.…”

Toward obtaining 2D and 3D and 1D PtPN with pentagonal pattern

“…Table II lists the side lengths (bond lengths) and angles (bond angles) forming the two distinct pentagons illustrated in Fig.2(a). Referring to the definitions for the 15 types of pentagons that can monohedrally tile a plane, 24 neither of the two types of pentagons in single-layer PtPN belongs to any of the 15 types. Therefore, the geometry of PtPN shows an example that a plane can still be tiled gaplessly by a combination of different types of pentagons from the 15 ones, retaining the anisotropy for a 2D material.…”

Constructing the 15th pentagon that tiles the plane

“…This arose from a systematic computer search described in their paper [1]. This arose from a systematic computer search described in their paper [1].…”