Polygonal Pieces Research Articles

Generative approaches for solving tangram puzzles

The Tangram is a dissection puzzle composed of seven polygonal pieces that can form different patterns. Solving the Tangram is an irregular shape packing problem known to be NP-hard. This paper investigates the application of four deep-learning architectures—Convolutional Autoencoder, Variational Autoencoder, U-Net, and Generative Adversarial Network—specifically designed for solving Tangram puzzles. We explore the potential of these architectures in learning the complex spatial relationships inherent in Tangram configurations. Our experiments show that the Generative Adversarial Network competes well with other architectures and converges considerably faster. We further prove that traditional evaluation metrics based on pixel accuracy often fail in assessing the visual quality of the generated Tangram solutions. We introduce a loss function based on a Weighted Mean Absolute Error that prioritizes pixels representing inter-piece sections over those covered by individual pieces. Extending this loss function, we propose a novel evaluation metric as a more fitting measure for assessing Tangram solutions compared to traditional metrics. This investigation advances our understanding of the capabilities of artificial intelligence in complex geometrical problem domains.

Economical Dissections

Abstract The Wallace-Bolyai-Gerwien theorem states any polygon can be decomposed into a finite number of polygonal pieces that can be translated and rotated to form any polygon of equal area. The theorem was proved in the early 19th century. The minimum number of pieces necessary to form these common dissections remains an open question. In 1905, Henry Dudney demonstrated a four-piece common dissection between a square and equilateral triangle. We investigate the possible existence of a three-piece common dissection. Specifically, we prove that there does not exist a three-piece common dissection using only convex polygons.

Exploring Geometry Teaching Model: Polygon Pieces and Dictionary Tools for the Model

Research reveals that many mathematics teachers find it difficult to stimulate learners' interest in learning geometry. One major reason suggested is that geometric concepts are not well conceptualised and comprehended by both learners and teachers. The study explored learners’ views on how polygon pieces and dictionary mediate learning of geometry. Nine Grade 8 learners were purposely selected from the cohort of 56 learners based on the diagnostic test results. By employing a qualitative approach through exploration data were gathered from semi- structured interviews and document analysis was implemented and reported in themes. The study found that polygon pieces with mathematics dictionary enhance learners’ learning of geometry through geometric inquiry. Polygon pieces assisted learners with geometric conceptualisation through cutting, constructing and measuring of angles and line segments. The dictionary enhanced learners’ geometric vocabulary by transferring informal vocabulary. We recommend mathematics teachers to integrate polygon pieces assisted by mathematics dictionary in the teaching and learning of mathematics.

Mixed integer quadratically-constrained programming model to solve the irregular strip packing problem with continuous rotations

The irregular strip packing problem consists of cutting a set of convex and non-convex two-dimensional polygonal pieces from a board with a fixed height and infinite length. Owing to the importance of this problem, a large number of mathematical models and solution methods have been proposed. However, only few papers consider that the pieces can be rotated at any angle in order to reduce the board length used. Furthermore, the solution methods proposed in the literature are mostly heuristic. This paper proposes a novel mixed integer quadratically-constrained programming model for the irregular strip packing problem considering continuous rotations for the pieces. In the model, the pieces are allocated on the board using a reference point and its allocation is given by the translation and rotation of the pieces. To reduce the number of symmetric solutions for the model, sets of symmetry-breaking constraints are proposed. Computational experiments were performed on the model with and without symmetry-breaking constraints, showing that symmetry elimination improves the quality of solutions found by the solution methods. Tests were performed with instances from the literature. For two instances, it was possible to compare the solutions with a previous model from the literature and show that the proposed model is able to obtain numerically accurate solutions in competitive computational times.

Computing and Fabricating Multiplanar Models

AbstractWe present a method for converting computer 3D models into physical equivalents. More specifically, we address the problem of approximating a 3D textured mesh using a small number of planar polygonal primitives that form a closed surface. This simplified representation allows us to easily manufacture individual components using computer controlled cutters (e.g., laser cutters or CNC machines). These polygonal pieces can be assembled into the final 3D model using internal planar connectors that are manufactured simultaneously. Our shape approximation algorithm iteratively assigns mesh faces to planar segments and slowly deforms these faces towards corresponding segments. This approach ensures that the output for a given closed mesh is still a closed mesh and avoids introducing self‐intersections. After this step we also compute the shape of polygonal connectors that internally hold the whole mesh surface. Both the polygonal surface elements and connectors can be manufactured in a single cutting pass. We validate the use of our method by computing and manufacturing a variety of textured polyhedral models.

Microstructure and Mechanical Performance of AZ31-1.7 Wt.% Si Alloy Processed by Cyclic Channel Die Compression

Cyclic channel die compression (CCDC) of AZ31-1.7 wt.% Si alloy was performed up to 5 passes at 623 K in order to investigate the microstructure and mechanical properties of compressed alloys. The results show that multi-pass CCDC is very effective to refine the matrix grain and Mg2Si phases. After the alloy is processed for 5 passes, the mean grain size decreases from 300 μm of as-cast to 8 μm. Both dendritic and Chinese script type Mg2Si phases break into small polygonal pieces and distribute uniformly in the matrix. The tensile strength increases prominently from 118 MPa to 216 MPa, whereas the hardness of alloy deformed 5 passes only increase by 8.4% compared with as-cast state.

A global constraint for nesting problems

Nesting problems are particularly hard combinatorial problems. They involve the positioning of a set of small arbitrarily-shaped pieces on a large stretch of material, without overlapping them. The problem constraints are bidimensional in nature and have to be imposed on each pair of pieces. This all-to-all pattern results in a quadratic number of constraints. Constraint programming has been proven applicable to this category of problems, particularly in what concerns exploring them to optimality. But it is not easy to get effective propagation of the bidimensional constraints represented via finite-domain variables. It is also not easy to achieve incrementality in the search for an improved solution: an available bound on the solution is not effective until very late in the positioning process. In the sequel of work on positioning non-convex polygonal pieces using a CLP model, this work is aimed at improving the expressiveness of constraints for this kind of problems and the effectiveness of their resolution using global constraints. A global constraint "outside" for the non-overlapping constraints at the core of nesting problems has been developed using the constraint programming interface provided by Sicstus Prolog. The global constraint has been applied together with a specialized backtracking mechanism to the resolution of instances of the problem where optimization by Integer Programming techniques is not considered viable. The use of a global constraint for nesting problems is also regarded as a first step in the direction of integrating Integer Programming techniques within a Constraint Programming model.

Hinged dissection of polyominoes and polyforms

A hinged dissection of a set of polygons S is a collection of polygonal pieces hinged together at vertices that can be rotated into any member of S. We present a hinged dissection of all edge-to-edge gluings of n congruent copies of a polygon P that join corresponding edges of P. This construction uses kn pieces, where k is the number of vertices of P. When P is a regular polygon, we show how to reduce the number of pieces to ⌈ k / 2 ⌉ ( n − 1 ) . In particular, we consider polyominoes (made up of unit squares), polyiamonds (made up of equilateral triangles), and polyhexes (made up of regular hexagons). We also give a hinged dissection of all polyabolos (made up of right isosceles triangles), which do not fall under the general result mentioned above. Finally, we show that if P can be hinged into Q, then any edge-to-edge gluing of n congruent copies of P can be hinged into any edge-to-edge gluing of n congruent copies of Q.

Dissecting Cuboids into Cuboids

1. INTRODUCTION. (Throughout this note we work in Euclidean space R.) Two convex polygons P and Q in the plane are said to be congruent by addition, written P Q, if we can dissect one of them into a finite number of polygonal jigsaw pieces that can be rearranged to form the second polygon. We denote a dissection of P by P = Pi + P2 + - - - + P,. By definition this means that the pieces P1, P2, ...., Pn are closed polygons whose union is P and that two distinct pieces have no interior points in common. Since the polygonal pieces can be cut into triangles, we can always assume that all pieces are triangles. Two polygons P and Q are said to be congruent by subtraction, denoted P - Q, if there exist polygons P' and Q' and polygonal dissections P' = P + P1 + - - - + Pn and Q' = Q + Q1 + - - - + Q, such that P', Pi,..., and P, are congruent to Q', Q1,..., and Q,, respectively. It is easy to see that congruence by addition and congruence by subtraction are equivalence relations. It is also clear that P Q implies P - Q, and P - Q implies that P and Q have the same area. Conversely, it is not difficult to show that P Q as soon as P and Q have the same area. Hence the two concepts of congruence by addition and subtraction are equivalent in the plane. A proof can be found in [3, Theorem 5.3.6]. One can readily generalize these notions to three dimensions. (In three dimensions one considers dissections into (irregular) tetrahedra.) A famous theorem of Dehn [2] asserts that a regular tetrahedron is not congruent by subtraction to a cube with the same volume, thus solving Hilbert's third problem. This result shows that it is impossible to devise a purely geometric theory of volume for three dimensional polytopes. We refer the reader to [3] and [5] for more information about Hilbert's problem and its history. 2. PROBLEM. We return to the plane and consider dissections of rectangles into rectangles instead of triangles. (In this note all rectangles are oriented so that their sides are parallel to the coordinate axes.) We consider the analogue of equivalence by subtraction that allows only rectangular pieces and denote this equivalence relation by =. For example, the rectangles in Figure 1 are equivalent.

A unique property of right-angled triangles

The familiar fact that we may cut any triangle into three polygonal pieces which can be rearranged to form a rectangle is one of the most elementary problems of polygonal dissection. This fact is shown in two ways in Figure 1 where both sides of the rectangle are predetermined after setting up the triangle: one side of the triangle and half of the altitude to this side, or one altitude and half of the side at right angles to it.

Polygon Area Decomposition for Multiple-Robot Workspace Division

A new polygon decomposition problem, the anchored area partition problem, which has applications to a multiple-robot terrain-covering problem is presented. This problem concerns dividing a given polygon P into n polygonal pieces, each of a specified area and each containing a certain point (site) on its boundary or in its interior. First the algorithm for the case when P is convex and contains no holes is presented. Then the generalized version that handles nonconvex and nonsimply connected polygons is presented. The algorithm uses sweep-line and divide-and-conquer techniques to construct the polygon partition. The input polygon P is assumed to have been divided into a set of p convex pieces (p = 1 when P is convex), which can be done in O(vP log log vP) time, where vP is the number of vertices of P and p = O(vP), using algorithms presented elsewhere in the literature. Assuming this convex decomposition, the running time of the algorithm presented here is O(pn2+vn), where v is the sum of the number of vertices of the convex pieces.

Hydatid cyst: Primary diagnosis by fine‐needle aspiration biopsy

Two cases of soft-tissues echinococcosis are presented in which the definitive diagnosis was established by identifying laminated layer remnants on fine-needle aspiration smears. Large polygonal pieces of acellular material with delicate parallel striations were seen, while neither scolices nor hooklets were found. The cases reported most likely represent the occurrence of sterile hydatid cysts. A diagnosis of echinococcosis should be considered when fragments suggestive of laminated layer are identified on smears, even in unusual sites and without evidence of hooklets and/or scolices.

The decomposition of polygons

In this paper a number of interesting results are defined and explained that are related to the notions of piecewise congruence by addition or subtraction. Two polygons are piecewise congruent by addition if either polygon can be cut into a finite number of polygonal pieces that can be rearranged to cover the other polygon. Initially, the fundamental theorem, that two polygons equal in area are also piecewise congruent by addition, is treated. Then it is shown, additionally, that there always exists a decomposition in which the sides of corresponding parts are parallel. Furthermore, for certain pairs of polygons, decompositions can be found which require only translations in relating corresponding parts. The necessary conditions for this theorem are found by using the boundary descriptions of the polygons under consideration.