Pucherite, BiV04, from Schneeberg, Saxony, is orthorhombic, space group Pnca, with a=5.328(2), b=5.052(2), c=12.003(3)A at 291 K, 2=4, Dx = 6.69 g em -3. The crystal structure of pucherite has been refined from single-crystal X-ray diffraction data to Rw = 0.020 for 221 unique reflections measured with AgKrx radiation. The structure consists of V04 tetrahedra and BiOs triangulated dodecahedra. ...

In a recent paper (arXiv:1505.01475 ) Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph G(10, 2), occurring as the...

Motivated by recent experiments on bilayer polyhedra composed of amphiphilic molecules, we study the elastic bending energies of bilayer vesicles forming polyhedral shapes. Allowing for segregation of excess amphiphiles along the ridges of polyhedra, we find that bilayer polyhedra can indeed have lower bending energies than spherical bilayer vesicles. However, our analysis also implies that, co...

The electronic structure of dodecahedral octacyanides of molybdenum IV and V is described in terms of SCCC molecular orbitals. Five MO's resemble d orbitals of the central atom. The splitting of appropriate levels is almost exactly the same as that following from the crystal field theory for GJG2 ~ 0.7. According to the theory stable Mo(CN)84" is dodecahedral and stable MO(CN)83is antiprismatic...

A finite subset of a Euclidean space is called an s-distance set if there exist exactly s values the distances between two distinct points in set. In this paper, we prove that maximum cardinality among all 5-distance sets $$\mathbb {R}^3$$ 20, and every with 20 similar to vertex regular dodecahedron.

We consider the Goldberg-Coxeter construction GCk,l(G0) (a generalization of a simplicial subdivision of the dodecahedron considered in [Gold37] and [Cox71]), which produces a plane graph from any 3or 4-valent plane graph for integer parameters k, l. A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face; a central circuit in ...

A particular U(N) gauge theory defined on the three dimensional dodecahedral lattice is shown to correspond to a model of oriented self-avoiding surfaces. Using large N reduction it is argued that the model is partially soluble in the planar limit. * Physique Théorique CNRS. ** Laboratoire de l’Institut de Recherche Fondamentale du Commissariat à l’Energie

It is shown that there exists a charge five monopole with octahedral symmetry and a charge seven monopole with icosahedral symmetry. A numerical implementation of the ADHMN construction is used to calculate the energy density of these monopoles and surfaces of constant energy density are displayed. The charge five and charge seven monopoles look like an octahedron and a dodecahedron respectivel...

In this paper we develop a new test to help identify whether a closed, orientable, irreducible 3–manifold is non-Haken. The test builds on work by Jaco and Oertel, and also incorporates heuristic pruning techniques to test whether a normal surface is compressible. As an application, we settle Thurston’s old question of whether the Weber-Seifert dodecahedral space is non-Haken. AMS Classificatio...