We consider vertex colourings of the dodecahedral graph with five colours, such that on each face vertices are coloured all colours. show total number these is 240. All can be obtained from any given colouring, by permuting and possibly applying central symmetry respect to centre regular dodecahedron. For colour classes form sets tetrahedra. These tetrahedra together one two compounds tetrahedr...

Porous materials have significant advantages, such as their light weight and good specific energy absorption. This paper presents the designs of two ordered Voronoi structures, a truncated octahedron (Octa) rhombic dodecahedron (Dodeca), based on spatial tessellation. Through numerical analysis, dynamic behavior, deformation absorption porous structures under different impact energies were expl...

In this work, copper–tungsten (Cu-W) composites with a cubic and rhombic dodecahedron W-skeleton were fabricated by the infiltration of Cu melt into three-dimensionally printed W scaffold. The effects skeleton structure on mechanical properties energy-absorbing characteristics Cu-W interpenetrating-phase composite investigated compared those commercial powder metallurgy. results indicated that ...

We report the results of a Monte Carlo study of the continuum limit of the two dimensional O(3) non-linear σ model. The notable finding is that it agrees very well with both the prediction inspired by Zamolodchikovs’ S-matrix ansatz and with the continuum limit of the dodecahedron spin model. The latter finding renders the existence of asymptotic freedom in the O(3) model rather unlikely. Asymp...

First, we prove a special case of Knaster’s problem, implying that each symmetric convex body in R3 admits an inscribed cube. We deduce it from a theorem in equivariant topology, which says that there is no S4-equivariant map from SO(3) to S2, where S4 acts on SO(3) as the rotation group of the cube and on S 2 as the symmetry group of the regular tetrahedron. We also give some generalizations. ...

Random packings containing 8192 hard spheres have been built with an ee-cient computer algorithm for various packing fractions up to c = 0:643, a value close to the upper limit c b ' 0:649. Long-range correlations and local order have been investigated via the calculation of the two-point correlation function g(r) and the Voronoi tessellation, respectively. The g(r) curve exhibits larger damped...

A class of two dinuclear dysprosium based complexes 1 and 2 were synthesized by employing salicyloylhydrazone derived pentadentate ligand (L). Structural analysis reveals that in complex 1, two DyIII centers are in muffin (Cs) coordination geometry while in 2, one DyIII center is in bicapped square antiprism (D4d) and other one is in triangular dodecahedron (D2d) coordination geometry. AC magne...

Local atomic arrangements were investigated as a model of amorphous iron in terms of Voronoi polyhedra for a dense random packing of hard spheres with and without relaxation. The structure of liquid iron was also simulated in a computer. As a result of relaxation, the frequency of five-edged faces increases at the expense of others, particularly the three-and seven-edged components. The amorpho...

The tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. From a first glance, one immediately notices that the Platonic Solids exhibit remarkable symmetry. They are the only convex polyhedra for which the same same regular polygon is used for each face, and the same number of faces meet at each vertex. Their symmetries are aesthetically pleasing, like those of stones cu...