We predict the three-dimensional structure of amorphous carbon generated by heating diamond superlattice at 6000 K with rapid quenching from liquid phase for densities ranging 2.0 to 3.5 g/cm3, in comparison 2.26 and 3.54 g/cm3 bulk graphite diamond, respectively. These predictions are based on reactive dynamics (RD) simulations using ReaxFF force field. Here, we simulate graphitization high te...

Candidates to the least perimeter partition of various polygonal shapes into N planar connected equal-area regions are calculated for N 6 42, compared to partitions of the disc, and discussed in the context of the energetic groundstate of a two-dimensional monodisperse foam. The total perimeter and the number of peripheral regions are presented, and the patterns classified according to the numb...

The complete symmetric directed graph of order $v$, denoted by $K_{v}$, is the on $v$~vertices that contains both arcs $(x,y)$ and $(y,x)$ for each pair distinct vertices $x$ and~$y$. For a given $D$, set all $v$ which $K_{v}$ admits $D$-decomposition called spectrum of~$D$-decomposition. There are 10 nonisomorphic orientations $7$-cycle (heptagon). In this paper, we completely settled problem ...

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...

Pentaheptites (three-coordinate tilings of the plane by pentagons and heptagons only) are classified under the chemically motivated restriction that all pentagons occur in isolated pairs and all heptagons have three heptagonal neighbors. They span a continuum between the two lattices exemplified by the boron nets in ThMoB4 (cmm) and YCrB4 (pgg), in analogy with the crossover from cubic-close-pa...

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...

Fused-pentagons results in an increase of local steric strain according to the isolated pentagon rule (IPR), and for all reported non-IPR clusterfullerenes multiple (two or three) metals are required to stabilize the strained fused-pentagons, making it difficult to access the single-atom properties. Herein, we report the syntheses and isolations of novel non-IPR mononuclear clusterfullerenes MN...

A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is known that a plane graph of minimum face size 5 contains light paths and a light pentagon. In this paper we show that every plane graph of minimum face size 5 contains also a light star K1,3 and we present a structural result concerning the existence of a pair of adjacent ...