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
This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18.
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 ℤ3 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.
In this paper we consider fullerene patches with nice boundaries containing between one and five pentagonal faces. We find necessary conditions for the side lengths of such patches, and then prove these conditions are sufficient by constructing such patches.
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