Tile Shapes Research Articles

비주기적 타일링을 이용한 융합교육 프로그램 개발 및 적용 연구

This study is a program development and application study to foster creativity and positive attitudes toward mathematics through tiling and mathematics. In school math, fourth graders in elementary school engage in regular pattern-making activities by pushing, flipping, and rotating flat shapes, and first graders in middle school cover interior angles of regular polygons. Practical Mathematics (2015 Curriculum), and Mathematics and Culture (2022 Curriculum), which are elective high school subjects, include tessellation activities. Tiling, also called tessellation, is commonly encountered in everyday life, such as in famous buildings like the Alhambra, as well as in home entrances, bathrooms, verandas, and squares. The works of artist Escher, who elevated such tiling into art, are widely known. Tiling is a convergence class subject which connects art and mathematics, and is a topic sufficient to stimulate students' curiosity and foster creativity. Recently there was a discovery of ‘aperiodic monotile’. Based on this discovery we raised following two questions: “We are planning to tile an infinitely wide bathroom floor. Are there tile shapes that can only tile the floor aperiodically?” and “We are planning to tile an infinitely wide bathroom floor. Is there a single tile shape that can only tile the floor aperiodically, when we only allow to use rotated versions and reflected versions of the tile?” Centered on these two questions, we developed a convergence education program and held a 10-session class for high school students. As a result, it was possible to see the possibility of increasing students' creativity and improving their positive attitude toward mathematics.

Creation of Dihedral Escher-like Tilings Based on As-Rigid-As-Possible Deformation

An Escher-like tiling is a tiling consisting of one or a few artistic shapes of tile. This article proposes a method for generating Escher-like tilings consisting of two distinct shapes (dihedral Escher-like tilings) that are as similar as possible to the two goal shapes specified by the user. This study is an extension of a previous study that successfully generated Escher-like tilings consisting of a single tile shape for a single goal shape. Building upon the previous study, our method attempts to exhaustively search for which parts of the discretized tile contours are adjacent to each other to form a tiling. For each configuration, two tile shapes are optimized to be similar to the given two goal shapes. By evaluating the similarity based on as-rigid-as possible deformation energy, the optimized tile shapes preserve the local structures of the goal shapes, even if substantial deformations are necessary to form a tiling. However, in the dihedral case, this approach is seemingly unrealistic because it suffers from the complexity of the energy function and the combinatorial explosion of the possible configurations. We developed a method to address these issues and show that the proposed algorithms can generate satisfactory dihedral Escher-like tilings in a realistic computation time, even for somewhat complex goal shapes.

Modeling the Interplay between Loop Tiling and Fusion in Optimizing Compilers Using Affine Relations

Loop tiling and fusion are two essential transformations in optimizing compilers to enhance the data locality of programs. Existing heuristics either perform loop tiling and fusion in a particular order, missing some of their profitable compositions, or execute ad-hoc implementations for domain-specific applications, calling for a generalized and systematic solution in optimizing compilers. In this article, we present a so-called basteln (an abbreviation for backward slicing of tiled loop nests) strategy in polyhedral compilation to better model the interplay between loop tiling and fusion. The basteln strategy first groups loop nests by preserving their parallelism/tilability and next performs rectangular/parallelogram tiling to the output groups that produce data consumed outside the considered program fragment. The memory footprints required by each tile are then computed, from which the upward exposed data are extracted to determine the tile shapes of the remaining fusion groups. Such a tiling mechanism can construct complex tile shapes imposed by the dependences between these groups, which are further merged by a post-tiling fusion algorithm for enhancing data locality without losing the parallelism/tilability of the output groups. The basteln strategy also takes into account the amount of redundant computations and the fusion of independent groups, exhibiting a general applicability. We integrate the basteln strategy into two optimizing compilers, with one a general-purpose optimizer and the other a domain-specific compiler for deploying deep learning models. The experiments are conducted on CPU, GPU, and a deep learning accelerator to demonstrate the effectiveness of the approach for a wide class of application domains, including deep learning, image processing, sparse matrix computation, and linear algebra. In particular, the basteln strategy achieves a mean speedup of 1.8× over cuBLAS/cuDNN and 1.1× over TVM on GPU when used to optimize deep learning models; it also outperforms PPCG and TVM by 11% and 20%, respectively, when generating code for the deep learning accelerator.

Nature favors simplicity and symmetry in biological forms

Why do symmetric and regular shapes abound in biology? Using arguments based on algorithmic information theory, we explain why a preference for simple shapes often occurs in biological systems. Our theory is validated via predicting the shape frequencies for naturally occurring protein assemblies, non-coding RNAs, self-assembled tile shapes, and a cell cycle mathematical model.

A Self-Replicating Single-Shape Tiling Technique for the Design of Highly Modular Planar Phased Arrays—The Case of L-Shaped Rep-Tiles

The design of irregular planar phased arrays (PAs) characterized by a highly-modular architecture is addressed. By exploiting the property of self-replicating tile shapes, also known as rep-tiles, the arising array layouts consist of tiles having different sizes, but equal shape, all being generated by assembling a finite number of smaller and congruent copies of a single elementary building-block. Towards this end, a deterministic optimization strategy is used so that the arising rep-tile arrangement of the planar PA is an optimal trade-off between complexity, costs, and fitting of user-defined requirements on the radiated power pattern, while guaranteeing the complete overlay of the array aperture. As a representative instance, such a synthesis method is applied to tile rectangular apertures with L-shaped tromino tiles. A set of representative results, concerned with ideal and real antenna models, as well, is reported for validation purposes, but also to point out the possibility/effectiveness of the proposed approach, unlike state-of-the-art tiling techniques, to reliably handle large-size array apertures.

Optimal Capacity-Driven Design of Aperiodic Clustered Phased Arrays for Multi-User MIMO Communication Systems

The optimal design of aperiodic/irregular clustered phased arrays (PAs) for base stations (BSs) in multi-user (UE) multiple-input–multiple-output (MU-MIMO) communication systems is addressed. This article proposes an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ad hoc</i> synthesis method aimed at maximizing the UEs traffic capacity within the cell served by the BS while guaranteeing a sufficient level of the signal at the terminals. Toward this end, the search of the optimal aperiodic clustering is carried out through a customized tiling technique able to consider both single and multiple tile shapes and assure the complete coverage of the antenna aperture for the maximization of the directivity. Representative results, from a wide set of numerical examples concerned with realistic antenna models and benchmark Third Generation Partnership Project (3GPP) scenarios, are reported to assess the advantages of the irregular array architectures in comparison with regular/periodic layouts proposed by the standard development organizations as well.

Improving the Air Permeability of Ventilated Roofs

In hot climates, ventilated pitched roofs help dissipating the excess heat, thus reducing the cooling energy demand. The key factor is the so-called Above Sheathing Ventilation (ASV) that depends on the air entering and leaving at the eaves, ridge and the gaps among tiles. A strategy for increasing the ASV is the enhancement of the roof air permeability through the development of new tile shapes, as proposed in the European Life HEROTILE project. The performance of a ventilated pitched roof are experimentally analysed by comparing some covering options: the new tile shape designs against the standard ones, available in the market, and a metal cover. A real scale pitched roof mock-up was built and equipped with a comprehensive monitoring system. The air flow and temperature in the different roof layers, as well as the heat flux passing through the roof and the cooling energy demand, were monitored according to the local weather conditions during the summer season. The analysis showed that the new tile design increases the ASV in comparison with the standard ones, and better thermal performance was achieved by reducing the heat gain due to solar radiation.

Families of convex tilings

We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile ``orientations'' determine the tiling. We associate to a tiling an underlying bipartite planar graph $G$ and its corresponding Kasteleyn matrix $K$. If $G$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences.

Escherization with Large Deformations Based on As-Rigid-As-Possible Shape Modeling

Escher tiling is well known as a tiling that consists of one or a few recognizable figures, such as animals. The Escherization problem involves finding the most similar shape to a given goal figure that can tile the plane. However, it is easy to imagine that there is no similar tile shape for complex goal shapes. This article devises a method for finding a satisfactory tile shape in such a situation. To obtain a satisfactory tile shape, the tile shape is generated by deforming the goal shape to a considerable extent while retaining the characteristics of the original shape. To achieve this, both goal and tile shapes are represented as triangular meshes to consider not only the contours but also the internal similarity of the shapes. To measure the naturalness of the deformation, energy functions based on traditional as-rigid-as-possible shape modeling are incorporated into a recently developed framework of the exhaustive search of the templates for the Escherization problem. The developed algorithms find satisfactory tile shapes with natural deformations for fairly complex goal shapes.

Optimization and Inverse Design of Floor Tile Airflow Distributions in Data Centers Using Response Surface Method

Abstract Computational fluid dynamics (CFD) has become a popular tool compared to experimental measurement for thermal management in data centers. However, it is very time-consuming and resource-intensive when used to model large-scale data centers, and often unrealistic for real-time thermal analyses. In addition, it is prohibitive to use CFD for the optimization process where thousands of designs need to be generated. Floor tile airflow distribution control is a key technique for maintaining a sufficient cold air delivery to variable thermal loadings of server cabinets. Regular practice of deploying a set of identical floor tiles may not result in the best solution for airflow uniformity through tiles. In this paper, an optimization procedure based on response surface methodology (RSM) is proposed to find the best arrangement of mixed-porosity floor tiles for different targeted tile airflow distributions. Fast-approximation RSM based on radial basis function (RBF) allows thousands of designs to be generated for the optimization process which uses a genetic algorithm as its main solver. The method shows proven success in maximizing floor tile airflow uniformity, and in the inverse design optimization where various tile airflow distribution topologies, i.e., linear, parabolic, and sinusoidal shapes are targeted. For the considered data center and aisle configuration, the improvement over the all-uniform-tile design is 55% in terms of standard deviation to the average tile airflow rate, whereas 90%, 91%, and 94% in root-mean-square error (RMSE) for the linear, parabolic, and sinusoidal floor tile airflow distribution objectives, respectively.

Monoparametric Tiling of Polyhedral Programs

Tiling is a crucial program transformation, adjusting the ops-to-bytes balance of codes to improve locality. Like parallelism, it can be applied at multiple levels. Allowing tile sizes to be symbolic parameters at compile time has many benefits, including efficient autotuning, and run-time adaptability to system variations. For polyhedral programs, parametric tiling in its full generality is known to be non-linear, breaking the mathematical closure properties of the polyhedral model. Most compilation tools therefore either perform fixed size tiling, or apply parametric tiling in only the final, code generation step. We introduce monoparametric tiling, a restricted parametric tiling transformation. We show that, despite being parametric, it retains the closure properties of the polyhedral model. We first prove that applying monoparametric partitioning (i) to a polyhedron yields a union of polyhedra with modulo conditions, and (ii) to an affine function produces a piecewise-affine function with modulo conditions. We then use these properties to show how to tile an entire polyhedral program. Our monoparametric tiling is general enough to handle tiles with arbitrary tile shapes that can tesselate the iteration space (e.g., hexagonal, trapezoidal, etc). This enables a wide range of polyhedral analyses and transformations to be applied.

An Efficient Exhaustive Search Algorithm for the Escherization Problem

In the Escherization problem, given a closed figure in the plane, the objective is to find a closed figure that is as close as possible to the input figure that tiles the plane. Koizumi and Sugihara’s formulation reduces this problem to an eigenvalue problem. In their formulation, only one of the potentially numerous templates is used to parameterize the possible tile shapes for each isohedral type. In this research, we try to search for the best tile shape using all possible templates for the nine most general isohedral types. This extension provides a considerable flexibility in possible tile shapes and improves the quality of the obtained tile shapes. However, the exhaustive search of all possible templates is computationally unrealistic using conventional calculation methods, and we develop an efficient algorithm to perform this in a reasonable computation time.

Flextended Tiles

Loop tiling to exploit data locality and parallelism plays an essential role in a variety of general-purpose and domain-specific compilers. Affine transformations in polyhedral frameworks implement classical forms of rectangular and parallelogram tiling, but these lead to pipelined start with rather inefficient wavefront parallelism. Multiple extensions to polyhedral compilers evaluated sophisticated shapes such as trapezoid or diamond tiles, enabling concurrent start along the axes of the iteration space; yet these resort to custom schedulers and code generators insufficiently integrated within the general framework. One of these modified shapes referred to as overlapped tiling also lacks a unifying framework to reason about its composition with affine transformations; this prevents its application in general-purpose loop-nest optimizers and the fair comparison with other techniques. We revisit overlapped tiling, recasting it as an affine transformation on schedule trees composable with any affine scheduling algorithm. We demonstrate how to derive tighter tile shapes with less redundant computations. Our method models the traditional “scalene trapezoid” shapes and novel “right-rectangle” variants. It goes beyond the state of the art by avoiding the restriction to a domain-specific language or introducing post-pass rescheduling and custom code generation. We conduct experiments on the PolyMage benchmarks and iterated stencils, validating the effectiveness and applicability of our technique on both general-purpose multicores and GPU accelerators.

Forming tile shapes with simple robots

Motivated by the problem of manipulating nanoscale materials, we investigate the problem of reconfiguring a set of tiles into certain shapes by robots with limited computational capabilities. As a first step towards developing a general framework for these problems, we consider the problem of rearranging a connected set of hexagonal tiles by a single deterministic finite automaton. After investigating some limitations of a single-robot system, we show that a feasible approach to build a particular shape is to first rearrange the tiles into an intermediate structure by performing very simple tile movements. We introduce three types of such intermediate structures, each having certain advantages and disadvantages. Each of these structures can be built in asymptotically optimal $$O(n^2)$$ rounds, where n is the number of tiles. As a proof of concept, we give an algorithm for reconfiguring a set of tiles into an equilateral triangle through one of the intermediate structures. Finally, we experimentally show that the algorithm for building the simplest of the three intermediate structures can be modified to be executed by multiple robots in a distributed manner, achieving an almost linear speedup in the case where the number of robots is reasonably small, and explain how the algorithm can be used to construct a triangle distributedly.

CFD analysis and experimental comparison of novel roof tile shapes

In tiled pitched roofs, a ventilated layer reduces the heat transfer between tiles and roof structure by means of natural and forced convection, thereby also reducing the cooling energy requirement. This effect could be enhanced by increasing the air permeability between the tiles by means of novel tile shapes, as proposed by the HEROTILE European project (LIFE14 CCA/IT/000939), of which this work presents the preliminary analysis supporting the new tile designs.Using an experimental rig, the air pressure difference and the volumetric flow rate between tiles have been measured for an existing Portoghese tile design over a range of pressures. Then, in order to understand the air flows under different conditions, a three-dimensional computational fluid dynamics (CFD) model has been implemented to recreate the full geometry of the rig. The model was calibrated against the aforementioned experimental results, and run with boundary conditions simulating different wind directions. Even in the low velocities typical of average local wind patterns, the fluid dynamic problem remains complex because of the geometry of the gaps between the tiles. However, it has been possible to assess the coefficient of local head loss and then apply it in an analytical relationship between pressure drop and flow rate, taking into account the open area. The results have shown how the wind direction affects the air permeability and, therefore, important insights have been gathered for the design of novel tiles.

CFD analysis of roof tile coverings

In hot climates tiled pitched roofs significantly reduce the heat transfer across the roof structure, due to the ventilated air layer between tiles and roofing underlay formed by the arrangement of battens and counter-battens supporting the tiles. This so-called Above Sheathing Ventilation (ASV) depends on the air entering and leaving at the eaves, ridge and the gaps between the tiles. With a view towards higher energy savings in space cooling, the natural and forced convection occurring in ASV could be enhanced by increasing the roof air permeability by means of novel tile shapes, as here analysed in two stages.The first stage of designing the new tile shapes was to measure the air permeability for a type of existing tile (Marseillaise style) using an experimental test rig, by monitoring the volumetric flow rate through the tiles over a range of pressure differences across the tiles. Then, a three-dimensional CFD model was implemented to replicate the full test rig geometry, and this was calibrated against the experimental data. In the next stage, the calibration was used to support the design of novel Marseillaise tile shapes, and to compare their performance against existing tiles. Finally, in order to analyse the variation in air flow under typical wind conditions for a pitched roof, a parametric study was undertaken, consisting of 72 scenarios varying wind speed, direction and angle of incidence.An increase in volumetric flow rate through the tiles was found to be related not only to an increase in the open area between tiles, but also to the design of the tile locks. By redesigning the geometry of these locks, whilst still giving consideration to their primary purpose of preventing the ingress of driving rain, it was possible to yield an improvement in air permeability of up to 100% in comparison with the original designs. Additionally, these novel designs were shown to increase the air flow rate as the wind angle moved from being directly up the roof slope around to the side, in contrast to the decrease seen with existing tile shapes.

Mechanical behavior of idealized, stingray-skeleton-inspired tiled composites as a function of geometry and material properties

Tilings are constructs of repeated shapes covering a surface, common in both manmade and natural structures, but in particular are a defining characteristic of shark and ray skeletons. In these fishes, cartilaginous skeletal elements are wrapped in a surface tessellation, comprised of polygonal mineralized tiles linked by flexible joints, an arrangement believed to provide both stiffness and flexibility. The aim of this research is to use two-dimensional analytical models to evaluate the mechanical performance of stingray skeleton-inspired tessellations, as a function of their material and structural parameters. To calculate the effective modulus of modeled composites, we subdivided tiles and their surrounding joint material into simple shapes, for which mechanical properties (i.e. effective modulus) could be estimated using a modification of traditional Rule of Mixtures equations, that either assume uniform strain (Voigt) or uniform stress (Reuss) across a loaded composite material. The properties of joints (thickness, Young’s modulus) and tiles (shape, area and Young’s modulus) were then altered, and the effects of these tessellation parameters on the effective modulus of whole tessellations were observed. We show that for all examined tile shapes (triangle, square and hexagon) composite stiffness increased as the width of the joints was decreased and/or the stiffness of the tiles was increased; this supports hypotheses that the narrow joints and high tile to joint stiffness ratio in shark and ray cartilage optimize composite tissue stiffness. Our models also indicate that, for simple, uniaxial loading, square tessellations are least sensitive and hexagon tessellations most sensitive to changes in model parameters, indicating that hexagon tessellations are the most “tunable” to specific mechanical properties. Our models provide useful estimates for the tensile and compressive properties of 2d tiled composites under uniaxial loading. These results lay groundwork for future studies into more complex (e.g. biological) loading scenarios and three dimensional structural parameters of biological tilings, while also providing insight into the mechanical roles of tessellations in general and improving the design of bioinspired materials.

Optimal Parallelogram Selection for Hierarchical Tiling

Loop tiling is an effective optimization to improve performance of multiply nested loops, which are the most time-consuming parts in many programs. Most massively parallel systems today are organized hierarchically, and different levels of the hierarchy differ in the organization of parallelism and the memory models they adopt. To make better use of these machines, it is clear that loop nests should be tiled hierarchically to fit the hierarchical organization of the machine; however, it is not so clear what should be the exact form of these hierarchical tiles. In particular, tile shape selection is of critical importance to expose parallelism of the tiled loop nests. Although loop tiling is a well-known optimization, not much is known about tile shape selection. In this article, we study tile shape selection when the shapes are any type of parallelograms and introduce a model to relate the tile shape of the hierarchy to the execution time. Using this model, we implement a system that automatically finds the tile shapes that minimize the execution time in a hierarchical system. Our experimental results show that in several cases, the tiles automatically selected by our system outperform the most intuitive tiling schemes usually adopted by programmers because of their simplicity.

The Relation Between Diamond Tiling and Hexagonal Tiling

Iterative stencil computations are important in scientific computing and more also in the embedded and mobile domain. Recent publications have shown that tiling schemes that ensure concurrent start provide efficient ways to execute these kernels. Diamond tiling and hybrid-hexagonal tiling are two tiling schemes that enable concurrent start. Both have different advantages: diamond tiling has been integrated in a general purpose optimization framework and uses a cost function to choose among tiling hyperplanes, whereas the greater flexibility with tile sizes for hybrid-hexagonal tiling has been exploited for effective generation of GPU code. In this paper we undertake a comparative study of these two tiling approaches and propose a hybrid approach that combines them. We analyze the effects of tile size and wavefront choices on tile-level parallelism, and formulate constraints for optimal diamond tile shapes. We then extend, for the case of two dimensions, the diamond tiling formulation into a hexagonal tiling one, which offers both the flexibility of hexagonal tiling and the generality of the original diamond tiling implementation. We also show how to compute tile sizes that maximize the compute-to-communication ratio, and apply this result to compare the best achievable ratio and the associated synchronization overhead for diamond and hexagonal tiling.

Processor Tile Shapes and Interconnect Topologies for Dense On-Chip Networks

We propose two eight-neighbor, two five-nearest-neighbor, and three six-nearest-neighbor interconnection topologies for many-core processor arrays-three of which use five-sided or hexagonal processor tiles-which typically reduce application communication distance and result in an overall application processor that requires fewer cores and lower power consumption. A 16-bit processor with the appropriate number of input and output ports is implemented in all topologies and tile shapes. The hexagonal and five-sided processor tiles and arrays of tiles are laid out with industry standard automatic place and route design flow and Manhattan-style wires without full-custom layout. A 1080p H.264/AVC residual video encoder and a 54 Mb/s 802.11a/g OFDM wireless local area network baseband receiver are mapped onto all topologies. The six-neighbor hexagonal tile incurs a 2.9% area increase per tile compared with the four-neighbor 2-D mesh, but its much more effective interprocessor interconnect yields an average total application area reduction of 22% and an average application power savings of 17%.