With the methods of barycentric coordinates, we establish several collinearities in the heptagonal triangle, formed by a side and two diagonals of different lengths of a regular heptagon. 1. The regular heptagon Consider a regular heptagon AA′C ′A′′BCB′ inscribed in a circle, each side of length a. The diagonals are of two kinds. Those with 3 vertices on the defining minor arc have the same len...

A heptagonal triangle is a non-isosceles triangle formed by three vertices of a regular heptagon. Its angles are π 7 , 2π 7 and 4π 7 . As such, there is a unique choice of a companion heptagonal triangle formed by three of the remaining four vertices. Given a heptagonal triangle, we display a number of interesting companion pairs of heptagonal triangles on its nine-point circle and Brocard circ...

We study the energy spectrum and persistent current of charge carriers confined in a graphene quantum ring geometry radius $R$ width $w$ subjected to magnetic flux. consider case where crystal symmetry is locally modified by replacing hexagon pentagon, square, heptagon or octagon. To model this type defect we include appropriate boundary conditions for angular coordinate. The electrons are fini...

Stone-Wales operators interchange four adjacent hexagons with two pentagon-heptagon 5|7 pairs that, graphically, may be iteratively propagated in the graphene layer, originating a new interesting structural defect called here Stone-Wales wave. By minimization, the Wiener index topological invariant evidences a marked anisotropy of the Stone-Wales defects that, topologically, are in fact prefera...

Helical metal-organic frameworks (MOFs) were used as templates or precursors to fabricate helical carbon nanorods (HCNRs) for the first time. contains many topological defects such pentagonal heptagonal carbons, which have potential facilitate oxygen reduction reactions (ORR). HCNRs show more positive onset/half-wave potentials and higher limited current density than straight (SCNRs). They also...

The purity of graphene samples is of crucial importance for their experimental and practical use. In this regard, the detection of the defects is of direct relevance. Here, we show that structural defects in graphene samples give rise to clear signals in the vibrational density of states (VDOS) at specific peaks at high and low frequencies. These can be used as an independent probe of the defec...