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You've probably noticed that when you blow a soap bubble, it quickly stretches into a nearly spherical shape.  But why should a bubble tend towards a spherical conformation, instead of a cube or an icosahedron?  According to physics, the bubble will tend toward a shape with the lowest energy, and the physical effect of surface tension is that this lowest energy is achieved when the bubble is in the shape with the least possible surface area enclosing a given volume.  For example, if we want to enclose one cubic inch of air, the smallest surface that could contain it is a perfect sphere.  The proof of this case, for a simple bubble, involves techniques from the mathematical study of [[differential geometry]], but turns out not to be too difficult.  Related problems in finding minimal surfaces, however, turn out to be quite fiendish, and involve computations of "mean curvature" related to the notions of curvature defined above.  The double bubble conjecture is one such.

You've probably noticed that when you blow a soap bubble, it quickly stretches into a nearly spherical shape.  But why should a bubble tend towards a spherical conformation, instead of a cube or an icosahedron?  According to physics, the bubble will tend toward a shape with the lowest energy, and the physical effect of surface tension is that this lowest energy is achieved when the bubble is in the shape with the least possible surface area enclosing a given volume.  For example, if we want to enclose one cubic inch of air, the smallest surface that could contain it is a perfect sphere.  The proof of this case, for a simple bubble, involves techniques from the mathematical study of [[differential geometry]], but turns out not to be too difficult.  Related problems in finding minimal surfaces, however, turn out to be quite fiendish, and involve computations of "mean curvature" related to the notions of curvature defined above.  The double bubble conjecture is one such.

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What is the smallest possible surface area for two bubbles joined stuck together, each enclosing some fixed volume?  Thinking back to childhood experience, we know such bubbles end up as two partial spheres, connected together at a perfectly flat surface.  But there are many other possibilities: why couldn't it be two tetrahedra, connected along some wavy surface?  Quite sophisticated mathematical techniques are needed to prove this seemingly obvious result.  The proof was not completed until 2000, by a team of mathematicians including [[User:RSchlafly|Roger Schlafly]].

What is the smallest possible surface area for two bubbles joined stuck together, each enclosing some fixed volume?  Thinking back to childhood experience, we know such bubbles end up as two partial spheres, connected together at a perfectly flat surface.  But there are many other possibilities: why couldn't it be two tetrahedra, connected along some wavy surface?  Quite sophisticated mathematical techniques are needed to prove this seemingly obvious result.  The proof was not completed until 2000, by a team of mathematicians including [[User:RSchlafly|Roger Schlafly]].