A three-dimensional pentagon-shaped stereo-tiling concept has been realized through use of penta-graphene carbon structures, although mathematically, regular planar pentagon shapes cannot be used to completely tile the Euclidean plane. Two applications of the findings of this study have been considered from the mathematical and engineering viewpoints. First, the proposed discovery facilitates math-rule-based generation of beautiful designs comprising star shapes formed using regular pentagons. The underlying mathematical logic and hexagon-division rules have been deduced to obtain the proposed pentagonal stereo-tiling pattern comprising equal-length bonds, and two aesthetic designs have been derived using the deduced logic. A fence-like structure exclusively comprising pentagram stars has been designed and given the name “Star Walls.” Emphasis has also been laid on application of the proposed stereo-tiling concept in industrial design operations, such as emboss manufacturing. To this end, finite element analysis of embossed steel sheets has been performed to verify the feasibility of the said industrial application.
Recent developments in computing technology and ever-increasing computational power have revolutionized modern-day applications of pure and applied mathematics. In 2015, for example, Bellos1 discovered a novel mathematically derived tile structure based on the pentagonal geometry. Around the same time, Dr. Mann et al.2 independently developed a monohedral tiling structure based on convex pentagons. The ancient Greek mathematician, Pythagoras, had demonstrated that only three shapes—the equilateral triangle, square, and regular hexagon—could be used to generate regular tiling patterns on a two-dimensional (2D) Euclidean plane. Extant studies performed on carbon-based materials, such as C60 fullerene,3–5 carbon nanotubes (CNTs),6,7 and graphene,8 have been based on the isolated pentagon rule (IPR), which essentially explains why none of the pentagons on the periphery of a soccer ball make contact with neighboring pentagons. IPR has especially been emphasized in the Nobel lecture delivered by Sir Harold W. Kroto.9 A regular dodecahedron is well recognized as a three-dimensional (3D) structure comprising twelve regular pentagons. However, Prinzbach et al.10 have reported that the lifetime of C20 fullerene is rather short.10 Another example of IPR violation is the C-66 fullerene structure encaging a scandium dimer (Sc2@C66), as reported in Ref. 11. In this structure, although the two pentagon pairs in C-66 are in contact with the other pentagon, Sc2@C66, as a whole, remains essentially stable.
Zhang et al.12 recently proposed a pentagraphene (PG) carbon structure (depicted in Figure 1(a)) suitable for use in stereo-tiling applications involving quasi-pentagonal geometries. Quasi-pentagons in the PG structures comprise five equal-length bonds and demonstrate a nonplanar arrangement.12 Stability of the PG structure has been confirmed via calculations based on the plane-wave density fluctuation theory (DFT). Results obtained from DFT simulations have predicted anomalies, such as negative Poisson ratio and increased stretchability, in the expansion behavior of PG carbon. In addition to these physical properties, the PG structure has been associated with the Cairo pentagonal tiling pattern (Figure 1(b)), which corresponds to projection of the PG structure onto a 2D plane. From the mathematical viewpoint, the relationship between PG structures and the Cairo pentagonal tiling pattern can be understood via standard realizations of 2-D crystal lattices using harmonic maps.13,14 The proposed study focuses on the mathematical interpretation of pentagonal stereo-tiling patterns to deduce novel patterns that can be created using the pentagon geometry.
As regards the relationship between fullerene and viruses, the use of sphere-like polyhedron structures with equal-length bonds, instead of sheet-like structures, such as PG carbon, has been discussed in Ref. 15. Convex polyhedrons with equal-length bonds and polyhedral symmetry can be classified into four types—the Platonic (regular) and Archimedean polyhedrons (such as the soccer ball) reported by ancient Greeks in the 17th century; rhombic polyhedral reported by Johannes Kepler; and lastly, Michael Goldberg (1937) proposed extensions to fullerene-based polyhedrons with non-planar quasi-hexagons, each of which possess six equal-length bonds.16 This last type of convex polyhedron is referred to as the “Goldberg polyhedron.” In this study, polyhedron geometries comprising nonplanar polygons have been considered as the mathematical backbone of stereo tiling applications involving quasi-pentagons.
II. RESULTS AND DISCUSSION
We studied pentagon-shaped stereo tiling employing the PG carbon structure. The mathematical rule of hexagon division to generate pentagons for stereo tiling has been proposed in the first subsection. Aesthetic stereo-tiling patterns obtained using pentagram stars are described in the second subsection. With regard to engineering applications, such as emboss manufacturing, based on the proposed concept, the mechanical stiffness of the PG structures has been evaluated via finite element analysis, as described in the last subsection.
A. Mathematical rule: Hexagon division to generate pentagons for stereo tiling
A unit cell of the Cairo pentagonal tiling, which corresponds to a projection of the PG carbon structure, can be considered as a tile comprising deformed hexagons. Figure 2(a) illustrates the rules to divide a single hexagon into multiple pentagons along with a schematic of the proposed structure. It must be noted that several rules exist to divide a structure into smaller pentagons. However, a consideration of all such rules is outside the scope of this study. The proposed study mainly focused on the generation of pentagons comprising equal-length bonds. To realize this objective, each end of a dividing line (red lines depicted in Figure 2) must either be connected to the opposite side of a regular hexagon or another dividing line. It has been observed that a hexagon comprising four pentagons (Figure 2(b)) is similar to the Cairo pentagon-tiling pattern (Figure 1(b)). For hexagons divided into two pentagons, the choice of a feasible tiling pattern becomes restricted to ribbon-like structure comprising hexagons. For hexagons divided into three pentagons, tiling with space, as depicted in Figure 2(c), can be made possible. Via generation of equal-length bonds by using DFT calculation as Carbon materials, the stereo tiling depicted in Figure 3 were obtained. Here, we performed structural optimization using Quantum Espresso.17 The structure of Figure 3(a) was the same as the PG structure shown in Figure 1. In Figure 3(b), we can observe small triangular formations consisting of the sides of two pentagons and one hexagon. This implies that this carbon contained within such triangle structures is not stable as Carbon materials. Although there does not exist another suitable carbon-based material, the stability of which in the proposed structure could be demonstrated in the proposed structure, the structure itself could be used to obtain beautiful industrial tiling designs, as depicted in Figure 4.
B. Application 1: Aesthetic design of stereo tiling of pentagram stars
Through generation of a 3D pentagonal structure, as depicted in Figure 3(a), the authors have been able to obtain a pure stereo-tiling pattern comprising pentagram stars, as depicted in Figure 4(b). Note that the concept of replacing equilateral-length pentagons with pentagram stars was explained in Figure 4(a). It must, however, be emphasized that the sheet-like structure depicted in the figure only comprises star patterns. From the viewpoint of design and aesthetics, the existence of stars within a tiling pattern appears rather interesting. In this study, the authors have deduced the logic to obtain as many quasi-pentagons as possible through use of convex polyhedrons. Yet another industrial design, depicted in Figure 4(c), could be generated using the deduced structure depicted in Figure 3(b). These patterns of Figures 4(b) and (c), with its fence-like structure, have been named “Star Walls” by the authors.
C. Application 2: Engineering stiffness of PG structures
Embossed metallic sheets, considered to be macroscopic systems comprising a PG structure, have also been considered in this study. These sheets possess high bending strength and rigidity, and form an effective lightweight material for use in the manufacture of industrial goods, such as automotive parts and construction materials. The Toyota Motor Corporation and others have recently investigated trial manufacturing of aluminum-alloy sheets with hexagonal dimple patterns (HDP),18 as depicted in Figure 5(a). The said HDP is similar to patterns obtained using PG carbon structures. Shapes of convex and concave components of the said HDP are depicted in Figure 5(b). Because experiments concerning the above-mentioned trial manufacturing were performed in 2004, equipment was not made available at the time of this study.19 Fortunately, however, the availability of finite element analysis (FEA) methods and 3D-printing technology enabled us to perform a similar trial manufacturing at low cost. To evaluate the stiffness of the emboss pattern, FEA analysis of a thin square-shaped cantilever plate was performed using the ADVENTURE software package.20 A single force of 3.0 N was considered to act at the free end of cantilever plate. Figure 5(c) depicts results obtained from FEA simulations that describe the magnitude of bending stiffness as a function of the depth of emboss patterns. Obtained results were found to be consistent with those obtained in trial investigations performed by Toyota.18 The proposed structure, which has been mathematically derived and comprises natural material, can therefore, be considered as an example that are aesthetic and excellent industrially.
Inspired by the PG carbon structure, the proposed study considers the design of a fence-like tiling pattern comprising an arrangement of quasi-pentagons in 3D space. In this study, bond lengths of the said quasi-pentagon were considered equal. Although quasi-pentagons are not flat, but rather bent, their parent structure is close to that of a regular pentagon. As a production rule of the PG carbon structure and its derivatives, the authors devised a means to divide a hexagon into constituent pentagons. By replacing the pentagon with star shapes, the authors have demonstrated the attainment of two aesthetic designs of stereo-tiling patterns, which have been named “Star Walls.” The proposed study also considers application of stereo tiling in the design of embossed structures. Through results of FEA simulations performed in this study, it has been demonstrated that the proposed stereo-tiling concept can serve as a useful, low-cost alternative in the design of stiff embossed structures.
This study was partially supported by the Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures (JHPCN) and the High Performance Computing Infrastructure (HPCI). The authors acknowledge Dr. S. H. Zhang, Dr. J. Zhou, Prof. Q. Wang, Prof. X. H. Chen, and Prof. P. Jena for the study of pentagraphene carbon as well as Dr. H. Naito, Prof. Y. Nishiura, Prof. M. Kotani, and Prof. J. Akiyama for mathematical discussions. The authors thank Mr. I. Kuroki, Mr. I. Takahashi, Dr. T. Takeda, Dr. R. Sahara, Prof. H. Tsutamori, Prof. M. Omiya, Prof. K. Ishii, Prof. S. Shimojo, Dr. S. Date, Dr. K. Yasufuku, and Prof. H. Takano for useful discussions, collaborations, and visualizations. This work was partially supported by JSPS KAKENHI, Japan, Grant no.: JP18H04494.
APPENDIX A: PROCEDURE FOLLOWED AND DETAILED RESULTS OBTAINED DURING FINITE ELEMENT ANALYSIS
The procedure followed and detailed results obtained during finite element analysis performed in this study (as summarized in Figure 5(c)) are described in this appendix. In this study, a square-shaped, thin cantilever-plate problem (Figure 6(a)) was solved to estimate the stiffness magnification caused by embossing. The said stiffness magnification was observed to depend on the angle θ of the arranged embossing pattern, where θ denotes the angle between the long axis of the concave pattern and the long axis of the original embossed pattern (Figure 6(b)). For the case depicted in Figure 6(c), θ = 45°. In the main text, only the case with θ = 0° has been presented. To facilitate systematic verification, detailed results of the FE analysis are presented here. Figure 6(d) depicts the finite-element mesh of the PG steel sheet with the original embossed pattern and 40% depth of embossing. The number of 10-noded tetrahedral elements for each model equaled approximately 0.4 million. The dependence of θ on stiffness magnification is depicted in Figure 6(e). For all examined values of θ, the stiffness magnification was confirmed to increase with increase in percentage depth of embossing. Figure 7 depicts the von Mises stress distribution of the PG steel sheet calculated using the said finite-element method for 0%, 20%, 40%, 60%, 80%, and 100% depths of embossing.