As starting point for patterns with seven-fold symmetry, we investigate the basic possibility to construct the regular heptagon by bicompasses and ruler. To cover the whole plane with elements of sevenfold symmetry is...As starting point for patterns with seven-fold symmetry, we investigate the basic possibility to construct the regular heptagon by bicompasses and ruler. To cover the whole plane with elements of sevenfold symmetry is only possible by overlaps and (or) gaps between the building stones. Resecting small parts of overlaps and filling gaps between the heptagons, one may come to simple parqueting with only a few kinds of basic tiles related to sevenfold symmetry. This is appropriate for parqueting with a center of seven-fold symmetry that is illustrated by figures. Choosing from the basic patterns with sevenfold symmetry small parts as elementary stripes or elementary cells, one may form by their discrete translation in one or two different directions periodic bordures or tessellation of the whole plane but the sevenfold point-group symmetry of the whole plane is then lost and there remains only such symmetry in small neighborhoods around one or more centers. From periodic tiling, we make the transition to aperiodic tiling of the plane. This is analogous to Penrose tiling which is mostly demonstrated with basic elements of fivefold symmetry and we show that this is also possible with elements of sevenfold symmetry. The two possible regular star-heptagons and a semi-regular star-heptagon play here a basic role.展开更多
We discuss a new possible construction of the regular heptagon by rhombic bicompasses explained in the text as a new geometric mean of constructions in the spirit of classical constructions in connection with an unmar...We discuss a new possible construction of the regular heptagon by rhombic bicompasses explained in the text as a new geometric mean of constructions in the spirit of classical constructions in connection with an unmarked ruler (straightedge). It avoids the disadvantages of the neusis construction which requires the trisection of an angle and which is not possible in classical way by compasses and ruler. The rhombic bicompasses allow to draw at once two circles around two fixed points in such correlated way that the position of one of the rotating points (arms) on one circle determines the position of the points on the other circle. This means that the positions of all points (arms) on both circles are determined in unique way.展开更多
In this paper the percolation behavior with a specific concentration of the defects was discussed on the twodimensional graphene lattice. The percolation threshold is determined by a numerical method with a high degre...In this paper the percolation behavior with a specific concentration of the defects was discussed on the twodimensional graphene lattice. The percolation threshold is determined by a numerical method with a high degree of accuracy. This method is also suitable for locating the percolation critical point on other crystalline structures. Through investigating the evolution of the largest cluster size and the cluster sizes distribution, we find that under various lattice sizes and concentrations of pentagon-heptagon defects there is no apparent change for the percolation properties in graphene lattice.展开更多
On some necessary conditions for double pyramidal central configurations with concave heptagon for any given ratio of masses, the existence and uniqueness of a class of double pyramidal central configurations with con...On some necessary conditions for double pyramidal central configurations with concave heptagon for any given ratio of masses, the existence and uniqueness of a class of double pyramidal central configurations with concave heptagon base for nine-body problems is proved in this paper, and the range of the ratio cr of the circularity radius of the heptagon to the half-height of the double pyramidal central configuration involved in this class configurations is obtained, which is in the interval （√3/3,1.099 600 679）, and the configuration involved in the bodies with any σ∈ （√3/3, 1.099 600 679） can form a central configuration which is a uniquely central configuration is proved.展开更多
Incorporating pentagons and heptagons into the hexagonal networks of pristine carbon nanotubes （CNTs） can form various CNT-based nanostructures, as pentagons and heptagons will bend or twist the CNTs by introducing ...Incorporating pentagons and heptagons into the hexagonal networks of pristine carbon nanotubes （CNTs） can form various CNT-based nanostructures, as pentagons and heptagons will bend or twist the CNTs by introducing positive and negative curvature, respectively. Some typical so-made CNT-based nanostructures are reviewed in this article, including zero-dimensional toroidal CNTs, and one-dimensional kinked and coiled CNTs. Due to the presence of non-hexagonal rings and curved geometries, such nanostructures possess rather different structural, physical and chemical properties from their pristine CNT counterparts, which are reviewed comprehensively in this article. Additionally, their synthesis, modelling studies, and potential applications are discussed.展开更多
Helical metal-organic frameworks(MOFs)were used as templates or precursors to fabricate helical carbon nanorods(HCNRs)for the first time.Helical carbon contains many topological defects such as pentagonal or heptagona...Helical metal-organic frameworks(MOFs)were used as templates or precursors to fabricate helical carbon nanorods(HCNRs)for the first time.Helical carbon contains many topological defects such as pentagonal or heptagonal carbons,which have the potential to facilitate oxygen reduction reactions(ORR).HCNRs show more positive onset/halfwave reduction potentials and higher limited current density than straight carbon nanorods(SCNRs).They also exhibit four-electron oxygen reduction in tests of ORR,while the alternative SCNRs prefer a two-electron reduction mechanism.Experimental and theoretical studies reveal that these enhanced ORR activities can be attributed to pentagon/heptagon defects in HCNRs.This work provides an effective strategy to synthesize helical,defect-rich carbon materials and opens up a new perspective for utilization of a spiral effect for the development of more effective electrocatalysts.展开更多
文摘As starting point for patterns with seven-fold symmetry, we investigate the basic possibility to construct the regular heptagon by bicompasses and ruler. To cover the whole plane with elements of sevenfold symmetry is only possible by overlaps and (or) gaps between the building stones. Resecting small parts of overlaps and filling gaps between the heptagons, one may come to simple parqueting with only a few kinds of basic tiles related to sevenfold symmetry. This is appropriate for parqueting with a center of seven-fold symmetry that is illustrated by figures. Choosing from the basic patterns with sevenfold symmetry small parts as elementary stripes or elementary cells, one may form by their discrete translation in one or two different directions periodic bordures or tessellation of the whole plane but the sevenfold point-group symmetry of the whole plane is then lost and there remains only such symmetry in small neighborhoods around one or more centers. From periodic tiling, we make the transition to aperiodic tiling of the plane. This is analogous to Penrose tiling which is mostly demonstrated with basic elements of fivefold symmetry and we show that this is also possible with elements of sevenfold symmetry. The two possible regular star-heptagons and a semi-regular star-heptagon play here a basic role.
文摘We discuss a new possible construction of the regular heptagon by rhombic bicompasses explained in the text as a new geometric mean of constructions in the spirit of classical constructions in connection with an unmarked ruler (straightedge). It avoids the disadvantages of the neusis construction which requires the trisection of an angle and which is not possible in classical way by compasses and ruler. The rhombic bicompasses allow to draw at once two circles around two fixed points in such correlated way that the position of one of the rotating points (arms) on one circle determines the position of the points on the other circle. This means that the positions of all points (arms) on both circles are determined in unique way.
文摘In this paper the percolation behavior with a specific concentration of the defects was discussed on the twodimensional graphene lattice. The percolation threshold is determined by a numerical method with a high degree of accuracy. This method is also suitable for locating the percolation critical point on other crystalline structures. Through investigating the evolution of the largest cluster size and the cluster sizes distribution, we find that under various lattice sizes and concentrations of pentagon-heptagon defects there is no apparent change for the percolation properties in graphene lattice.
基金Funded by NSF (Natural Science Foundation) of China (No. 10231010) and NSF of Chongqing Educational Committee (KJ051109, KJ06110X), NSF of Chongqing Science and Technology Committee, NSF of CQSXXY
文摘On some necessary conditions for double pyramidal central configurations with concave heptagon for any given ratio of masses, the existence and uniqueness of a class of double pyramidal central configurations with concave heptagon base for nine-body problems is proved in this paper, and the range of the ratio cr of the circularity radius of the heptagon to the half-height of the double pyramidal central configuration involved in this class configurations is obtained, which is in the interval （√3/3,1.099 600 679）, and the configuration involved in the bodies with any σ∈ （√3/3, 1.099 600 679） can form a central configuration which is a uniquely central configuration is proved.
文摘Incorporating pentagons and heptagons into the hexagonal networks of pristine carbon nanotubes （CNTs） can form various CNT-based nanostructures, as pentagons and heptagons will bend or twist the CNTs by introducing positive and negative curvature, respectively. Some typical so-made CNT-based nanostructures are reviewed in this article, including zero-dimensional toroidal CNTs, and one-dimensional kinked and coiled CNTs. Due to the presence of non-hexagonal rings and curved geometries, such nanostructures possess rather different structural, physical and chemical properties from their pristine CNT counterparts, which are reviewed comprehensively in this article. Additionally, their synthesis, modelling studies, and potential applications are discussed.
基金financial support from the National Key Research and Development Program of China(2018YFA0208600 and 2017YFA0700100)the Key Research Program of Frontier Science,CAS(QYZDJ-SSW-SLH045)+2 种基金the National Natural Science Foundation of China(21671188,21871263 and 22033008)the Strategic Priority Research Program of CAS(XDB20000000)the Youth Innovation Promotion Association,CAS(2014265)。
文摘Helical metal-organic frameworks(MOFs)were used as templates or precursors to fabricate helical carbon nanorods(HCNRs)for the first time.Helical carbon contains many topological defects such as pentagonal or heptagonal carbons,which have the potential to facilitate oxygen reduction reactions(ORR).HCNRs show more positive onset/halfwave reduction potentials and higher limited current density than straight carbon nanorods(SCNRs).They also exhibit four-electron oxygen reduction in tests of ORR,while the alternative SCNRs prefer a two-electron reduction mechanism.Experimental and theoretical studies reveal that these enhanced ORR activities can be attributed to pentagon/heptagon defects in HCNRs.This work provides an effective strategy to synthesize helical,defect-rich carbon materials and opens up a new perspective for utilization of a spiral effect for the development of more effective electrocatalysts.
