The PM3 quantum-mechanical method is able to model the magic water clusters (H20),, and (H20)&+. Results indicate that the H30+ ion is tightly bound within the (H20),, cluster by multiple hydrogen bonds, causing deformation to the symmetric (HzO),, pentagonal dodecahedron structure. The structures, energetics, and hydrogen bond patterns of six local minima (H20)21H+ clusters are presented.

With this study, the Android Studio development environment and Java language in three-dimensional space, It can be easily applied to user with graphics a reliable solution number of combinations.  is aimed create digital lock program. Graphics Rotation: This application running on devices has been written successfully screen The corners regular dodecahedron placed middle are indicated dif...

Let P be the right-angled hyperbolic dodecahedron or 120-cell, and let W be the group generated by reflections across codimension-one faces of P . We prove that if Γ ⊂ W is a torsion free subgroup of minimal index, then the corresponding hyperbolic manifold Hn/Γ is determined up to homeomorphism by Γ modulo the symmetry group of P .

We develop a combinatorial method to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle. Platonic graphs with this property are called topologically uniquely Hamiltonian. The same method is used to demonstrate topologically distinct Hamiltonian cycles on the icosahedron graph and to show that a regular graph embeddable on the 2-holed torus is topolog...

In many scientific and engineering applications, there are occasions where points need to be inserted uniformly onto a sphere. Previous works on uniform point insertion mainly focus on the offline version, i.e., to compute N positions on the sphere for a given interger N with the objective to distribute these points as uniformly as possible. An example application is the Thomson problem where t...

The IMO Jury looked for a relatively simple geometric question for the IMO paper, and decided to use only the planar version of the problem. The answer in the planar variant is not surprising: any completely symmetric set consists of the vertices of a regular polygon. The straightforward generalization to space would read: a completely symmetric set consists either of the vertices of a regular ...