The tangram and Sei Shōnagon Chie no Ita are popular dissection puzzles consisting of seven pieces. Each puzzle can be formed by identifying edges from sixteen identical right isosceles triangles. It is known that the tangram can form 13 convex polygons. We show that Sei Shōnagon Chie no Ita can form 16 convex polygons, propose a new puzzle that can form 19, no 7 piece puzzle can form 20, and 1...

in this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. we focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of fomin and reading, and colored quiver mutation.

In four experiments we investigated whether human observers are able to use certain regularities in polygons when they have to indicate which polygon they perceive as the more regular in a paired comparison task. From our results we conclude that regularities that are restricted to the contour do not play a role in judgments of regularity. For instance, equilateral polygons are not considered t...

In this paper, we precise the asymptotic behaviour of Newton polygons of L-functions associated to character sums, coming from certain n variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum ∆ = ∆1 ⊕ ∆2 when we know the limit of generic Newton polygons for each facto...

We consider orthogonal polygons with vertices located at integer lattice points. We show that if all of the sides of a simple orthogonal polygon without holes have odd lengths, then it cannot be tiled by dominoes. We provide similar characterizations for orthogonal polygons with sides of arbitrary length. We also give some generalizations for polygons with holes and polytopes in 3 dimensions.

Spatial models are often based on polygons both in 2D and 3D. Many Geo-ICT products support spatial data types, such as the polygon, based on the OpenGIS ‘Simple Features Specification’. OpenGIS and ISO have agreed to harmonize their specifications and standards. In this paper we discuss the relevant aspects related to polygons in these standards and compare several implementations. A quite exh...

TheMinkowski sum of two setsA,B ∈ IR, denotedA⊕B, is defined as {a+ b | a ∈ A, b ∈ B}. We describe an efficient and robust implementation for the construction of Minkowski sums of polygons in IR using the convolution of the polygon boundaries. This method allows for faster computation of the sum of non-convex polygons in comparison to the widely-used methods for Minkowski-sum computation that d...

We consider the problem of partitioning rectilinear polygons into rectangles, using segments of minimum total length. This problem is NP-hard for polygons with holes. Even for hole-free polygons no known algorithm can nd an optimal partitioning in less than O(n 4) time. We present the rst linear-time algorithm for computing rectangula-tions of hole-free polygons, within a constant factor of the...

In this paper a proof of the Jordan curve theorem will be presented. Some familiarity with the basic notions of nonstandard analysis is assumed. The rest of the paper is selfcontained except for some standard theorems about polygons. The theorem will be proved in what ought to be a natural way: by approximation by polygons. This method is not usually found in the standard proofs since the appro...

We study 4 × 4 games for which the best response dynamics contain a cycle. We give examples in which multiple Shapley polygons occur for these kinds of games. We derive conditions under which Shapley polygons exist and conditions for the stability of these polygons. It turns out that there is a very strong connection between the stability of heteroclinic cycles for the replicator equation and S...