Deformations of Interfaces

We must keep in mind that a surface, apparently smooth at low resolution, may reveal complicated structure under higher resolutions. At all spatial scales surfaces are like coastlines; surfaces appear bumpy at different scales. Objects whose magnified pieces are similar or look similar to the whole are self-similar. The lengths we measure depend on the lengths of our rulers. Scaling means that the properties of what's observed depend on the scale we use to measure them.

We can show how the area of any surface in three dimensions may increase if we dimple the surface so that no surface can become a local maximum for its area functional. Critical points other than a dimpled maximum or minimum are saddle points of the area functional. Minimal surfaces at saddle points are unstable, so even small fluctuations in amplitude may lead to collapse. Think of blowing on a soap bubble. The bubble indents at one location but bulges out at another, but the volume of the bubble remains unchanged unless our breath warms the bubble. Now, given a hemocoel at its smallest dimensions in which the hemcoel behaves as a two-dimensional sheet, assume its infinite planar surface to be minimal. Now a closed curve anywhere on this minimal surface will have the smallest possible area having the curve as its boundary.

Before continuing, we must keep in mind that the mathematics of surfaces suggest simple physical "solutions" that nature never realizes completely, because real surfaces and fluids are never entirely "homogeneous." Asymmetrical forces and chaos always intervene to perturb surfaces and systems that approach equilibrium (Meakin, 1998).

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