Geometry and mechanics have both a relevant role in determining the three-dimensional packing of $ 8 bubbles displayed similar foam structure. We assume that spatial arrangement obeys geometrical principle maximizing minimum mutual distance between bubble centroids. The compacted structure is then obtained by radially under constraint volume conservation. generate polygonal tiling on central sp...

In this account recent progress in enhancing the complexity of liquid crystal self-assembly is highlighted. The discussed superstructures are formed mainly by polyphilic T-shaped and X-shaped molecules composed of a rod-like core, tethered with glycerol units at both ends and flexible non-polar chain(s) in lateral position, but also related inverted molecular structures are considered. A series...

We address the question of single flip discrete dynamics in sets of two-dimensional random rhombus tilings with fixed polygonal boundaries. Single flips are local rearrangements of tiles which enable to sample the configuration sets of tilings via Markov chains. We determine the convergence rates of these dynamical processes towards the statistical equilibrium distributions and we demonstrate t...

There is only one finite, 2-connected, linearly convex graph in the Archimedean triangular tiling that does not have a Hamiltonian cycle. The vertices and polygonal edges of the planar Archimedean tilings 44 and 36 of the plane, partially shown in Figs. 1 and 2, respectively, are called the square tiling graph (STG) and the triangular tiling graph (TTG). (See [1].) A subgraph G of TTG is linear...

The salient feature of one-cell-thick epithelia is their en face view, which reveals the polygonal cross section of the close-packed prismatic cells. The physical mechanisms that shape these tissues were hitherto explored using theories based on cell proliferation, which were either entirely topological or included certain morphogenetic forces. But mitosis itself may not be instrumental in mold...