The study of the minimum Born–Oppenheimer structures of the protonated water clusters, H2O nH, is performed for n=20 and 21. The structures belonging to four basic morphologies are optimized at the Hartree–Fock, second-order many-body perturbation theory and coupled cluster level, with the 6–31G, 6-31G , and 6-311G basis sets, using the parallel ACES III program. The lowest energy structure for...

In 1857, the Irish mathematician Sir William Hamilton(1805-1865) invented a game of travelling around the edges of a graph from vertex to vertex. Hamilton described the game, in a letter to his friend Graves, as a mathematical game on the dodecahedron in which one person sticks five pins in any five consecutive vertices and the other is required to complete the path to form a spanning cycle. In...

Proteins that self-assemble into regular shell-like polyhedra are useful, both in nature and in the laboratory, as molecular containers. Here we describe cryo-electron microscopy (EM) structures of two versatile encapsulation systems that exploit engineered electrostatic interactions for cargo loading. We show that increasing the number of negative charges on the lumenal surface of lumazine syn...

Clathrin triskelions form polyhedral cages with hexagonal and pentagonal faces when dialyzed against suitable assembly buffers. However, when the buffer is made 12% saturated in ammonium sulfate and the dialysis is performed at 4 degrees C, clathrin polymerizes into cubes. The cube is constructed from eight triskelions with one at each corner. The edge length of the cube is approximately 45 nm,...

Motivated by the relation between particle shape and packing, we measure the volume fraction ϕ occupied by the Platonic solids which are a class of polyhedrons with congruent sides, vertices, and dihedral angles. Tetrahedron-, cube-, octahedron-, dodecahedron-, and icosahedron-shaped plastic dice were fluidized or mechanically vibrated to find stable random loose packing ϕ(rlp)=0.51,0.54,0.52,0...

a w o 3 bstract leven sealers, including Resilon sealer, were mixed ccording to manufacturer’s instructions. Setting times ere determined in both aerobic and anaerobic envionments. Two samples of each sealer were mixed and laced in the following conditions: (a) uncovered in an erobic incubator; (b) covered with a glass cover-slip in he aerobic incubator; (c) covered with phosphate buffred salin...

What is the shape of the Universe? Is it curved or flat, finite or infinite ? Is space “wrapped around” to create ghost images of faraway cosmic sources? We review how tessellations allow to build multiply-connected 3D Riemannian spaces useful for cosmology. We discuss more particularly the proposal of a finite, positively curved, dodecahedral space for explaining some puzzling features of the ...

Motivated by modern applications like image processing and wireless sensor networks, we consider a variation of the Kepler Conjecture. Given any infinite set of unit balls covering the whole space, we want to know the optimal (lim sup) density of the volume which is covered by exactly one ball (i.e., the maximum such density over all unit ball covers, called the optimal 1-density and denoted as...

In this paper we survey results on packings of congruent spheres in 3-dimensional spaces of constant curvature. The topics discussed are as follows: Hadwiger numbers of convex bodies and kissing numbers of spheres; Touching numbers of convex bodies; Newton numbers of convex bodies; One-sided Hadwiger and kissing numbers; Contact graphs of finite packings and the combinatorial Kepler problem; Is...