Minimal cones on hypercubes

It is shown that in dimension greater than 4, the minimal area hypersurface separating the faces of a hypercube is the cone over the edges of the hypercube. This constrasts with the cases of two and three dimensions, where the cone is not minimal. For example, a soap film on a cubical frame has a small rounded square in the center. In dimensions over 6, the cone is minimal even if the area separating opposite faces is given zero weight. The proof uses the maximal flow problem that is dual to the minimal surface problem

Instability of the wet cube cone soap film

A dry conical soap film on a cubical frame is well known not to be stable. Recent experimental evidence seems to indicate that adding liquid to form Plateau borders stabilizes the conical film, perhaps to arbitrarily low liquid volumes. This paper presents numerical simulation evidence that the wet cone is unstable for low enough liquid volume, with the critical volume fraction being about 0.000274

The Surface Evolver

The Surface Evolver is a computer program that minimizes the energy of a surface subject to constraints. The surface is represented as a simplicial complex. The energy can include surface tension, gravity, and other forms. Constraints can be geometrical constraints on vertex positions or constraints on integrated quantities such as body volumes. The minimization is done by evolving the surface down the energy gradient. This paper describes the mathematical model used and the operations available to interactively modify the surface

Tensor virial equation of evolving surfaces in sintering of aggregates of particles by diffusion

The moment of inertia tensor is a quantity that characterizes the morphology of aggregates of particles. The deviatoric components indicate the anisotropy of the aggregate, and its compactness is described by the isotropic component, i.e. the second moment of inertia, which is related to the radius of gyration. The equation of motion of the moment of inertia tensor is proposed for the sintering and coalescence of crystalline particles by bulk diffusion and surface diffusion. Simulations of the evolution of aggregates of particles (linear chains, rings and branched chains) show that the aggregates become more compact and more isotropic structures, driven by the surface energy tensor or the surface force density. The tensor virial equation for diffusion is applicable also to evolution of pores, precipitates and inclusions embedded in a surrounding matrix

Aggregates of two-dimensional vesicles: Rouleaux and sheets

Using both numerical and variational minimization of the bending and adhesion energy of two-dimensional lipid vesicles, we study their aggregation, and we find that the stable aggregates include an infinite number of vesicles and that they arrange either in a columnar or in a sheet-like structure. We calculate the stability diagram and we discuss the modes of transformation between the two types of aggregates, showing that they include disintegration as well as intercalation.Comment: 4 figure

Soap films and covering spaces

A new mathematical model of soap films is proposed, called the covering space model. The two sides of a film are modelled as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities, films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have proved troublesome for previous soap film models

The Opaque Cube Problem

It is a classic puzzle to find the shortest set of curves that intersect all straight lines through a square, and the conjectured solution is still unproven. This paper asks the analogous question for a cube, and comes up with the best known solution

Computation of equilibrium foam structure using the Surface Evolver

The Surface Evolver has been used to minimise the surface area of various ordered structures for monodisperse foam. Additional features have enabled its application to foams of arbitrary liquid fraction. Early results for the case of dry foam (negligible liquid fraction) produced a structure haveing lower surface area, or energy, than Kelvin\u27s 1887 minimal tetrakaidecahedron. The calculations reported here show that this remains the case when the liquid fraction is finite, up to about 11%, at which point an f.c.c arrangement of the cells becomes preferable

Numerical Solution of Soap Film Dual Problems

The soap film problem is to minimize area, and its dual is to maximize the flux of a divergenceless bounded vectorfield. This paper discretizes the continuous problem and solves it numerically. This gives upper and lower bounds on the area of the globally minimizing film. In favorable cases, the method can be used to discover previously unknown films. No initial assumptions about the topology of the film are needed. The paired calibration or covering space model of soap films is used to enable representation of films with singularities

Minimal Surfaces, Corners, and Wires

Weierstrass representations are given for minimal surfaces that have free boundaries on two planes that meet at an arbitrary dihedral angle. The contact angles of a surface on the planes may be different. These surfaces illustrate the behavior of soapfilms in convex and nonconvex corners. They can also be used to show how a boundary wire can penetrate a soapfilm with a free end, as in the overhand knot surface. They should also cast light on the behavior of capillary surfaces