Phase transition creates the geometry of the continuum from discrete space.

Models of discrete space and space-time that exhibit continuum-like behavior at large lengths could have profound implications for physics. They may help tame the infinities arising from quantizing gravity, and remove the need for the machinery of the real numbers; a construct with no direct observational support. However, despite many attempts to build discrete space, researchers have failed to produce even the simplest geometries.

Flow and evolution of ice-sucrose crystal mushes.

We study the rheology of suspensions of ice crystals at moderate to high volume fractions in a sucrose solution in which they are partially soluble, a model system for a wide class of crystal mushes or slurries. Under step changes in shear rate, the viscosity changes to a relaxed value over several minutes, in a manner well fitted by a single exponential. The behavior of the relaxed viscosity is power-law shear thinning with shear rate, with an exponent of -1.

Optimal counter-current exchange networks.

We present a general analysis of exchange devices linking their efficiency to the geometry of the exchange surface and supply network. For certain parameter ranges, we show that the optimal exchanger consists of densely packed pipes which can span a thin sheet of large area (an "active layer"), which may be crumpled into a fractal surface and supplied with a fractal network of pipes. We derive the efficiencies of such exchangers, showing the potential for significant gains compared to regular exchangers (where the active layer is flat), using parameters relevant to biological systems.

Simple heuristic for the viscosity of polydisperse hard spheres.

We build on the work of Mooney [Colloids Sci. 6, 162 (1951)] to obtain an heuristic analytic approximation to the viscosity of a suspension any size distribution of hard spheres in a Newtonian solvent. The result agrees reasonably well with rheological data on monodispserse and bidisperse hard spheres, and also provides an approximation to the random close packing fraction of polydisperse spheres.

Easily repairable networks: reconnecting nodes after damage.

We introduce a simple class of distribution networks that withstand damage by being repairable instead of redundant. Instead of asking how hard it is to disconnect nodes through damage, we ask how easy it is to reconnect nodes after damage. We prove that optimal networks on regular lattices have an expected cost of reconnection proportional to the lattice length, and that such networks have exactly three levels of structural hierarchy.

Close packing density of polydisperse hard spheres.

The most efficient way to pack equally sized spheres isotropically in three dimensions is known as the random close packed state, which provides a starting point for many approximations in physics and engineering. However, the particle size distribution of a real granular material is never monodisperse. Here we present a simple but accurate approximation for the random close packing density of hard spheres of any size distribution based upon a mapping onto a one-dimensional problem.

Sintering behavior of two roughened crystals just after contact.

We consider two spherical, roughened crystals with approximately isotropic surface free energy, which are brought into contact and begin to sinter. We argue that the geometry immediately postcontact is two dimensional and Cartesian and can be approximated by the evolution of a slot-shaped cavity. On this basis, we construct traveling wave solutions for the crystal shape in the limits of bulk- and surface-diffusion-limited kinetics.

Fractal design for an efficient shell strut under gentle compressive loading.

Because of Euler buckling, a simple strut of length L and Young modulus Y requires a volume of material proportional to L3f12} in order to support a compressive force F, where f=F/YL2 and f<<1. By taking into account both Euler and local buckling, we provide a hierarchical design for such a strut consisting of intersecting curved shells, which requires a volume of material proportional to the much smaller quantity L3f exp[2 square root(ln 3)(ln f-1)].

Fractal design for efficient brittle plates under gentle pressure loading.

We consider a plate made from an isotropic but brittle elastic material, which is used to span a rigid aperture, across which a small pressure difference is applied. The problem we address is to find the structure which uses the least amount of material without breaking. Under a simple set of physical approximations and for a certain region of the pressure-brittleness parameter space, we find that a fractal structure in which the plate consists of thicker spars supporting thinner spars in an hierarchical arrangement gives a design of high mechanical efficiency.