The Graphs of Planar Soap Bubbles

We characterize the graphs formed by two-dimensional soap bubbles as being exactly the 3-regular bridgeless planar multigraphs. Our characterization combines a local characterization of soap bubble graphs in terms of the curvatures of arcs meeting at common vertices, a proof that this characterization remains invariant under Moebius transformations, an application of Moebius invariance to prove bridgelessness, and a Moebius-invariant power diagram of circles previously developed by the author for its applications in graph drawing.Comment: 16 pages, 9 figure

Influences on the formation and evolution of Physarum polycephalum inspired emergent transport networks

The single-celled organism Physarum polycephalum efficiently constructs and minimises dynamical nutrient transport networks resembling proximity graphs in the Toussaint hierarchy. We present a particle model which collectively approximates the behaviour of Physarum. We demonstrate spontaneous transport network formation and complex network evolution using the model and show that the model collectively exhibits quasi-physical emergent properties, allowing it to be considered as a virtual computing material. This material is used as an unconventional method to approximate spatially represented geometry problems by representing network nodes as nutrient sources. We demonstrate three different methods for the construction, evolution and minimisation of Physarum-like transport networks which approximate Steiner trees, relative neighbourhood graphs, convex hulls and concave hulls. We extend the model to adapt population size in response to nutrient availability and show how network evolution is dependent on relative node position (specifically inter-node angle), sensor scaling and nutrient concentration. We track network evolution using a real-time method to record transport network topology in response to global differences in nutrient concentration. We show how Steiner nodes are utilised at low nutrient concentrations whereas direct connections to nutrients are favoured when nutrient concentration is high. The results suggest that the foraging and minimising behaviour of Physarum-like transport networks reflect complex interplay between nutrient concentration, nutrient location, maximising foraging area coverage and minimising transport distance. The properties and behaviour of the synthetic virtual plasmodium may be useful in future physical instances of distributed unconventional computing devices, and may also provide clues to the generation of emergent computation behaviour by Physarum. © Springer Science+Business Media B.V. 2010

Multispherical shapes of vesicles highlight the curvature elasticity of biomembranes

Giant lipid vesicles form unusual multispherical or “multi-balloon” shapes consisting of several spheres that are connected by membrane necks. Such multispherical shapes have been recently observed when the two sides of the membranes were exposed to different sugar solutions. This sugar asymmetry induced a spontaneous curvature, the sign of which could be reversed by swapping the interior with the exterior solution. Here, previous studies of multispherical shapes are reviewed and extended to develop a comprehensive theory for these shapes. Each multisphere consists of large and small spheres, characterized by two radii, the large-sphere radius, Rl, and the small-sphere radius, Rs. For positive spontaneous curvature, the multisphere can be built up from variable numbers Nl and Ns of large and small spheres. In addition, multispheres consisting of N*=Nl+Ns equally sized spheres are also possible and provide examples for constant-mean-curvature surfaces. For negative spontaneous curvature, all multispheres consist of one large sphere that encloses a variable number Ns of small spheres. These general features of multispheres arise from two basic properties of curvature elasticity: the local shape equation for spherical membrane segments and the stability conditions for closed membrane necks. In addition, the (Nl+Ns)-multispheres can form several (Nl+Ns)-patterns that differ in the way, in which the spheres are mutually connected. These patterns may involve multispherical junctions consisting of individual spheres that are connected to more than two neighboring spheres. The geometry of the multispheres is governed by two polynomial equations which imply that (Nl+Ns)-multispheres can only be formed within a certain restricted range of vesicle volumes. Each (Nl+Ns)-pattern can be characterized by a certain stability regime that depends both on the stability of the closed necks and on the multispherical geometry. Interesting and challenging topics for future studies include the response of multispheres to locally applied external forces, membrane fusion between spheres to create multispherical shapes of higher-genus topology, and the enlarged morphological complexity of multispheres arising from lipid phase separation and intramembrane domains