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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 ? 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. Keywords: tiling, pentagon
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An $N$-tiling of triangle $ABC$ by triangle $T$ (the `tile') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(\alpha,\beta,\gamma)$, and sides $(a,b,c)$. This paper takes up the case when $3\alpha + 2\beta = \pi$. Then there are (as was already known) exactly five possible shapes of $ABC$: either $ABC$ is isosceles with base angles $\alpha$, $\beta$, or $\alpha+\beta$, or the angles of $ABC$ are $(2\alpha,\beta,\alpha+\beta)$, or the angles of $ABC$ are $(2\alpha, \alpha, 2\beta)$. In each of these cases, we have discovered, and here exhibit, a family of previously unknown tilings. These are tilings that, as far as we know, have never been seen before. We also discovered, in each of the cases, a Diophantine equation involving $N$ and the (necessarily rational) number $s = a/c$ that has solutions if there is a tiling using tile $T$ of some $ABC$ not similar to $T$. By means of these Diophantine equations, some conclusions about the possible values of $N$ are drawn; in particular there are no tilings possible for values of $N$ of certain forms. We prove, for example, that there is no $N$-tiling with $N$ prime when $3\alpha + 2\beta = \pi$. These equations also imply that for each $N$, there is a finite set of possibilities for the tile $(a,b,c)$ and the triangle $ABC$. (Usually, but not always, there is just one possible tile.) These equations provide necessary, and in three of the five cases sufficient, conditions for the existence of $N$-tilings.
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An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N trianglescongruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we study the case of isosceles (but not equilateral) ABC. There are three possible forms of the tile: right-angled, or with one angle double another, or with a 120 degree angle. We study all three cases. In the case of a right-angled tile, we give a complete characterization of the tilings. In the latter two cases we prove the ratios of the sides of the tile are rational, and give a necessary condition for the existence of an N-tiling. For the case when the tile has one angle double another, we prove N cannot be prime.
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European Journal of Combinatorics
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