The mechanism of translation symmetry breakdown in newly proposed low-dimensional carbon pentagon-constituted nanostructures (e.g., pentagraphene) with multiple sp(2)/sp(3) sublattices was studied by GGA DFT, DFTB, and model potential approaches. It was found that finite nanoclusters suffer strong uniform unit cell bending followed by breaking of crystalline lattice linear translation invarianc...

In this paper, we study the number of tilings of the hyperbolic plane that can be constructed, starting from a single pentagonal tile, the only permitted transformations on the basic tile being the replication by displacement along the lines of the pentagrid. We obtain that there is no such tiling with five colours, that there are exactly two of them with four colours and a single trivial tilin...

Tensegrity structures are spatial reticulated structures composed of cables and struts. Tensegrity systems are good candidates for adaptive and deployable structures and thus have applications in various engineering fields. A “hollow-rope” tensegrity system composed of tensegrity-ring modules has been demonstrated by the authors to be a viable system for a pedestrian bridge. This paper focuses ...

We present scanning tunnelling microscopy studies and first principles calculation on the 2D crystallization of pentagonal pentamethylcorannulene on a Cu(111) surface under ultrahigh vacuum in the temperature range of 50 K to 400 K. The observed 2D crystal phases and their packing densities are compared to tiling options of hard pentagons. Temperature change-induced reversible phase transitions...

I discuss the symmetry of fullerenes, viruses and geodesic domes within a unified framework of icosadeltahedral representation of these objects. The icosadeltahedral symmetry is explained in details by examination of all of these structures. Using Euler’s theorem on polyhedra, it is shown how to calculate the number of vertices, edges, and faces in domes, and number of atoms, bonds and pentagon...

Supramolecular organization of a metal complex may significantly contribute to the magnetization dynamics of mononuclear SMMs. This is illustrated for a heptacoordinated Fe(II) complex with rather moderate Ising-type anisotropy for which a slow magnetization relaxation with significant energy barrier was reached when this complex was properly organized in the crystal lattice. Incidentally, it i...

A fullerene is any molecule composed entirely of carbon, in the form of a hollow sphere, ellipsoid, or tube. Spherical fullerenes are also called buckyballs, and they resemble the balls used in Association Football. Cylindrical ones are called carbon nanotubes or buckytubes. Fullerenes are similar in structure to graphite, which is composed of stacked graphene sheets of linked hexagonal rings; ...

The deliberate synthesis of the Keplerate [K(20) subset{(W)W(5)O(21)(SO(4))}(12)(VO)(30)(SO(4))(H(2)O)(63)](18-) with 20 pores all closed by K(+) in a supramolecular fashion proves that it is possible to follow new routes in polyoxotungstate chemistry based on pentagonal {(W)W(5)}-type units and to tune magnetic exchange couplings in {(M)M(5)}(12)M'(30) type Keplerates.

Polygonal gold nanoplates are generated in situ in poly(vinyl alcohol) film through thermal treatment, the polymer serving as the reducing agent and stabilizer for the nanoparticle formation and enforcing preferential orientation of the plates. The rare pentagonal as well as the more commonly observed hexagonal, triangular and square/rectangle shapes are obtained by fine-tuning the Au/PVA ratio...

In a recent paper, I began with four classical results due to Jacobi, Dirichlet and Lorenz which give the number of representations of a number by various forms involving squares in terms of divisor functions. I deduced another sixteen similar representation theorems involving triangular numbers and/or squares, including celebrated results of Legendre and Ramanujan (and omitted a seventeenth, g...