Read the abstract, see who wrote it and who cites it, and get the citation in APA, MLA, Chicago or Harvard.
About 359 papers for “Hexagonal tiling”
Z. A. Zhu, Erxiao Wang, Min Yan · ArXiv.org · 2025
Since the thesis of K. Reinhardt in 1918, it is well known that there are exactly three types of convex hexagons that can tile the plane. However, the proof of the fact is far from being complete. We prove this fact, under an assumption much weaker than the convexity.
Cited by 0Open access
École Normale, Sven Verdoolaege, Albert Cohen et al. · 2016
state.edu Iterative stencil computations are important in scientific com-puting and more and more also in the embedded and mobile domain. Recent publications have shown that tiling schemes that ensure concurrent start provide efficient ways to exe-cute these kernels. Diamond tiling and hybrid-hexagonal tiling are two successful tiling schemes that enable concur-rent start. Both have different advantages: diamond tiling is integrated in a general purpose optimization framework and uses a cost function to choose among tiling hyperplanes, whereas the more flexible tile sizes of hybrid-hexagonal tiling have proven to be effective for the generation of GPU code. We show that these two approaches are even more in-teresting when combined. We revisit the formalization of diamond and hexagonal tiling, present the effects of tile size and wavefront choices on tile-level parallelism, and formulate constraints for optimal diamond tile shapes. We then extend the diamond tiling formulation into a hexagonal tiling one, combining the benefits of both. The paper closes with an outlook of hexagonal tiling in higher dimensional spaces, an important generalization suitable for massively parallel ar-chitectures. 1.
Cited by 0
John C. Baez · Notices of the American Mathematical Society · 2024
Cited by 0Open access
Tobias Grosser, Albert Cohen, Justin Holewinski et al. · 2014
Time-tiling is necessary for the efficient execution of iterative stencil computations. Classical hyper-rectangular tiles cannot be used due to the combination of backward and forward dependences along space dimensions. Existing techniques trade temporal data reuse for inefficiencies in other areas, such as load imbalance, redundant computations, or increased control flow overhead, therefore making it challenging for use with GPUs.
Cited by 66
Laurinda C Brown, Ronald L. Ives · Bristol Research (University of Bristol) · 1989
Cited by 0
MingGao LI · Open Physics · 2026
Abstract A lattice gas model with short-range interactions is considered in this paper. Through Monte Carlo simulations, it was found that the system shows glassy relaxation behaviors. Moreover, an exotic hexagonal tiling order emerges at low temperatures. Similar real-space configurations have been observed in an alloy before. The novel order also brings about novel percolations. The order and the percolations have profound effects on the dynamics of the system and are responsible for the dynamic heterogeneity.
Cited by 0Open access
Jaya Abraham, Dersana mbika K.S · International Journal of Mathematics Trends and Technology · 2018
Cited by 0
ISOKAWA, Yukinao, 135612 · Kagoshima University Repository · 2020
Many strapworks seen in medieval Islamic arts can be made by tessellating several kinds of girih tiles. The present paper studies tilings with the hexagonal girih tile. First we show that there exists a unique tiling with five-fold rotational symmetry. Next tilings that contain tree-like sub-patterns are introduced. Remarkably then can contain at most two tree-like sub-patterns. Finally all periodic tilings are determined. We show further that a similar way of making such periodic tilings can generate uncountably many aperiodic tilings.
Cited by 0Open access
Grasselli, Luigi · IRIS UNIMORE (University of Modena and Reggio Emilia) · 2021
Cited by 0Open access
Hong Li (20183), Xiao-Rui Ren (12055952), Yan Wang (15435) et al. · Figshare · 2025
Reticular chemistry has greatly stimulated the development of framework materials, yet covalent organic frameworks (COFs) featuring irregular tiling are rare, because of the strict constraints on irregular tile lengths and angles. Guided by reticular chemistry, we deconstructed type III irregular hexagonal tiling into a combination of <i>D</i><sub>2<i>h</i></sub> + <i>D</i><sub>2<i>h</i></sub> monomers and achieved the first successful synthesis of 2D COFs featuring such tiling with <b>hit</b> topology. By tuning the crystal growth conditions, we obtained a set of reticular isomers, COF-hit and COF-bex, with distinct reticular structures. Powder X-ray diffraction and high-resolution transmission electron microscopy were employed to precisely characterize their topology structures. COF-hit demonstrates outstanding water absorption capabilities under high humidity conditions. This rational design of COFs using irregular tiling opens a new avenue to diversify the structural types and topological varieties of COFs and promote the development of reticular chemistry.
Cited by 0Open access
Giudiceandrea Federico, Grasselli Luigi · Iris Unimore (University of Modena and Reggio Emilia) · 2021
Generalizing Fermat and Napoleon points of a triangle, we introduce the notion of complementary Jacobi points, showing their collinearity with the circumcenter of the given triangle. The coincidence of the associated perspective lines for complementary Jacobi points is also proved, together with the orthogonality of this line with the one joining the circumcenter and the Jacobi points. Involutions on the Kiepert hyperbola naturally arise, allowing a geometric insight on the relationship between Jacobi points, their associated perspective lines and Kiepert conics of a triangle.
Cited by 0Open access
Kohei Iritani, Motoki Ikeda, Anna Yang et al. · Langmuir · 2017
We present here hexagonal tiling using hexagonal phenylene-ethynylene and phenylene-butadiynylene macrocycles attached by alkyl ester groups, PEM-C6 and PBM-C8, respectively, or triethylene glycol ester groups, PEM-TEG and PBM-TEG, respectively, at each vertex of the macrocyclic periphery at the liquid/solid interface. In this study, we focused on the effects of macrocyclic core size and the chemical properties of side chains attached to macrocyclic cores as well as solute concentrations on the hexagonal geometry of self-assembled monolayers. STM observations at the 1,2,4-trichrolobenzene/graphite interface revealed that PEM-C6 formed a honeycomb structure by van der Waals interactions between the interdigitated alkyl chains. However, upon increasing solute concentration, it changed to more dense hexagonal structure (tentatively called loose hexagonal structure I). In contrast, PBM-C8 formed loose hexagonal structure II of a slightly different packing mode at low concentration, while at high concentration it formed a high-density hexagonal structure in which alkyl chains are not adsorbed on the surface (dense hexagonal structure). In the dense hexagonal structure, macrocyclic cores are linked by hydrogen bonds between the ester carbonyl oxygen and the aromatic hydrogen atoms of the neighboring macrocycles. The packing geometries of loose hexagonal structures of PEM-C6 and PBM-C8 are different due to the different distance between the attachment of the alkyl ester groups which are located in confined space. On the other hand, PEM-TEG and PBM-TEG formed dense hexagonal structures, similar to PBM-C8 at high concentration, with their TEG units not adsorbed on the surface.
Cited by 25
Kohei Iritani (1402825), Motoki Ikeda (4513972), Anna Yang (1493869) et al. · Figshare · 2017
We present here hexagonal\ntiling using hexagonal phenylene-ethynylene\nand phenylene-butadiynylene macrocycles attached by alkyl ester groups, <b>PEM-C6</b> and <b>PBM-C8</b>, respectively, or triethylene\nglycol ester groups, <b>PEM-TEG</b> and <b>PBM-TEG</b>, respectively, at each vertex of the macrocyclic periphery at the\nliquid/solid interface. In this study, we focused on the effects of\nmacrocyclic core size and the chemical properties of side chains attached\nto macrocyclic cores as well as solute concentrations on the hexagonal\ngeometry of self-assembled monolayers. STM observations at the 1,2,4-trichrolobenzene/graphite\ninterface revealed that <b>PEM-C6</b> formed a honeycomb structure\nby van der Waals interactions between the interdigitated alkyl chains.\nHowever, upon increasing solute concentration, it changed to more\ndense hexagonal structure (tentatively called loose hexagonal structure\nI). In contrast, <b>PBM-C8</b> formed loose hexagonal structure\nII of a slightly different packing mode at low concentration, while\nat high concentration it formed a high-density hexagonal structure\nin which alkyl chains are not adsorbed on the surface (dense hexagonal\nstructure). In the dense hexagonal structure, macrocyclic cores are\nlinked by hydrogen bonds between the ester carbonyl oxygen and the\naromatic hydrogen atoms of the neighboring macrocycles. The packing\ngeometries of loose hexagonal structures of <b>PEM-C6</b> and <b>PBM-C8</b> are different due to the different distance between\nthe attachment of the alkyl ester groups which are located in confined\nspace. On the other hand, <b>PEM-TEG</b> and <b>PBM-TEG</b> formed dense hexagonal structures, similar to <b>PBM-C8</b> at high concentration, with their TEG units not adsorbed on the\nsurface.
Cited by 0Open access
Jinyu Wang, Yifei Kang, Yiwen Li et al. · 2021
Nowadays, multiple Field Programmable Gate Arrays (FPGAs) accelerators have been widely used in stencil computation fields. However, the state-of-the-art hexagonal tiling algorithm that efficiently improves stencil computation performance is mainly designed for CPUs or GPUs, which not suitable to directly process on FPGAs, leading to lower performance. To address this, a hexagonal tiling based multiple FPGAs stencil computation architecture and the corresponding optimization algorithm are proposed in this paper. The architecture uses the on-chip registers to store and carry cells data of a hexagonal tile. In this way, the scale and size of tiles are dramatically increased as well as the intra-FPGA calculation performance. Then, to take full advantage of multiple FPGAs processing ability, a memory shared inter-FPGAs high speed data transfer structure is devised. Finally, the Mixed-Integer Linear Programming (MILP) is used to optimize an objective function which considers the candidate FPGAs costs, computation latency and resources utilization to obtain a desirable tile size and layout result. The proposed method has been validated on the FPGA cluster which consists of two Xilinx Alveo U50 and one Alveo U250 devices. And experimental results show that we achieve performance up to 580 Gflop/s using one U50 device and 2261 Gflop/s using three FPGAs. The proposed optimizer is also tested with state-of-the-art multiple FPGAs stencil computation acceleration method and the performance is increased by 21.8% at most and 20.04% on average.
Cited by 0
Habib M. Ammari · 2019
Coverage is an essential task in the design of wireless sensor networks. We noticed that the problem of coverage in a two-dimensional (2D) space has similarity with the 2D tiling problem, which can be stated as follows: How can a 2D space be tiled by replicas of a set (or tiles)? This is a 2D instance of the second part of Hilbert's 18th problem: Is there a polyhedron that admits an anisohedral tiling only in three dimensions, i.e., tiles a 3D space, but does not admit an isohedral tiling? In this paper, we investigate the problem of 2D k-coverage, where each point in a 2D field is covered by at least k sensors, k≥1. In our study, we found that it is helpful to identify a 2D convex tile that best approximates the sensors' sensing range. First, we propose some sensor placement strategies using a hexagonal tiling-based method, and compute the corresponding sensor density. Second, we suggest a more general one using irregular hexagon, denoted by IRH(r/n), where r stands for the radius of the sensors' sensing range, and n≥2. We show that IRH(r/n) is a 2D tile, and derive the corresponding minimum sensor density. Third, we compute the relationship between the sensors' communication range R and r for each placement strategy. We corroborate our analysis with simulation results.
Cited by 7
Kai Li, Xiao‐Rui Ren, Yan Wang et al. · Journal of the American Chemical Society · 2025
Reticular chemistry has greatly stimulated the development of framework materials, yet covalent organic frameworks (COFs) featuring irregular tiling are rare, because of the strict constraints on irregular tile lengths and angles. Guided by reticular chemistry, we deconstructed type III irregular hexagonal tiling into a combination of <i>D</i><sub>2<i>h</i></sub> + <i>D</i><sub>2<i>h</i></sub> monomers and achieved the first successful synthesis of 2D COFs featuring such tiling with <b>hit</b> topology. By tuning the crystal growth conditions, we obtained a set of reticular isomers, COF-hit and COF-bex, with distinct reticular structures. Powder X-ray diffraction and high-resolution transmission electron microscopy were employed to precisely characterize their topology structures. COF-hit demonstrates outstanding water absorption capabilities under high humidity conditions. This rational design of COFs using irregular tiling opens a new avenue to diversify the structural types and topological varieties of COFs and promote the development of reticular chemistry.
Cited by 7
Akihisa Koga, Sam Coates · Physical review. B./Physical review. B · 2022
We study magnetic properties of the half-filled Hubbard model on the two-dimensional hexagonal golden-mean tiling. We find that the vertex model of the tiling is bipartite, with a sublattice imbalance of $\sqrt{5}/(6{\ensuremath{\tau}}^{3})$ (where $\ensuremath{\tau}$ is the golden mean), and that the noninteracting tight-binding model gives macroscopically degenerate states at $E=0$. We clarify that each sublattice has specific types of confined states, which in turn leads to an interesting spatial pattern in the local magnetizations in the weak coupling regime. Furthermore, this allows us to analytically obtain the lower bound on the fraction of the confined states as $(\ensuremath{\tau}+9)/(6{\ensuremath{\tau}}^{6})\ensuremath{\sim}0.0986$, which is conjectured to be the exact fraction. These results imply that a ferrimagnetically ordered state is realized even in the weak coupling limit. The introduction of the Coulomb interaction lifts the macroscopic degeneracy at the Fermi level and induces finite staggered magnetization as well as uniform magnetization. Likewise, the spatial distribution of the magnetizations continuously changes with increasing interaction strength. The crossover behavior in the magnetically ordered states is also addressed in terms of the perpendicular space analysis.
Cited by 19Open access
古賀昌久, Akihisa Koga, Sam Coates · Tokyo Tech Research Repository (Tokyo Institute of Technology) · 2022
Cited by 0
T. Kamaraj, D. G. Thomas · 2014
Regional hexagonal Tile rewriting grammars(RHTRG) are the recently introduced hexagonal picture generating devices which used a simple type of tiling called regional hexagonal tiling. This model is having isometric rules of derivation and more general than context-free Hexagonal array grammars (HAG) but incomparable with hexagonal tiling systems (HTS). In this paper we compare RHTRG with some parallel generating formalisms like Extended pure 2D hexagonal context-free grammars with regular control and hexagonal array token Petri net Structure with respect to generating capacity..
Cited by 0
Koga, Akihisa, Coates, Sam · arXiv (Cornell University) · 2022
We study magnetic properties of the half-filled Hubbard model on the two-dimensional hexagonal golden-mean tiling. We find that the vertex model of the tiling is bipartite, with a sublattice imbalance of $\sqrt{5}/(6τ^3)$ (where $τ$ is the golden mean), and that the non-interacting tight-binding model gives macroscopically degenerate states at $E=0$. We clarify that each sublattice has specific types of confined states, which in turn leads to an interesting spatial pattern in the local magnetizations in the weak coupling regime. Furthermore, this allows us to analytically obtain the lower bound on the fraction of the confined states as $(τ+9)/(6τ^6)\sim 0.0986$, which is conjectured to be the exact fraction. These results imply that a ferrimagnetically ordered state is realized even in the weak coupling limit. The introduction of the Coulomb interaction lifts the macroscopic degeneracy at the Fermi level, and induces finite staggered magnetization as well as uniform magnetization. Likewise, the spatial distribution of the magnetizations continuously changes with increasing interaction strength. The crossover behavior in the magnetically ordered states is also addressed in terms of the perpendicular space analysis.
Cited by 0Open access