Area Minimization Problem
The basic problem is to minimize length, area, or some other quantity, among curves or surfaces satisfying a given constraint. Some well-known examples are that a circle has least perimeter among curves enclosing a given area, a square has least perimeter among rectangles with the same area, and a sphere has least area among surfaces enclosing the same volume. Geometry provides a great playground full of optimization problems. Lawlor has helped develop the method known ``mapped slicing'' which provides proofs for various minimization problems. Students may work on proving that subsets of the octahedral star are area-minimizing. The octahedral star W is a soap film whose boundary is formed by the edges of a regular octahedron. It is piecewise-planar, formed as the union of twelve isosceles triangles and six kites. A picture can be viewed at http://www.susqu.edu/brakke/evolver/examples/octa/octafilm.htm , with the label "Octabest.fe." Ken Brakke has conjectured that this surface is the least-area surface on this boundary, with the requirement that competing surfaces must topologically separate the solid octahedron into components, with the eight faces all being separated from one another. Four other soap films are known to satisfy these requirements, but have been calculated by Brakke to have more surface area than W. The other four films are not piecewise-planar. If a surface is area-minimizing then its subsets are also area-minimizing; otherwise one could replace a piece with something of less area and thus obtain a global surface with less area than the original. Thus, while working toward a proof of Brakke's conjecture, one might try to first prove that interesting pieces of the octahedral star W are themselves area-minimizing. In fact, any intersection of W with a very small ball is area-minimizing, since such pieces are simply planar triple junctions or tetrahedral cones. We have a method called mapped calibration that is showing much promise of late on other minimization questions. We have known for some time that mapped calibration provides a new minimization proof for tetrahedral cones, and for (at least slightly) larger pieces of the octahedral star, containing two or more tetrahedral singularity points. Now would be a good time for undergraduates to investigate how big of pieces of W can be proved minimizing by mapped calibration. Students would certainly get some results, and possibly even solve the minimization question for all of W.
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