Dr. Gary Lawlor

Geometric Optimization

The basic problem in geometric optimization is to minimize length, area, or some other quantity, among curves or surfaces satisfying a given constraint. A well-known example is that a circle has least perimeter among curves enclosing a given area. Some problems that the REU participants could research are: (1) the octahedron problem which asks what is the least area surface spanning the edges of the octahedron. The conjectured solution is a beautiful, piecewise-planar soap film consisting of twelve triangles and six kites; (2) large minimizing networks in the hyperbolic plane; that is finding the shortest path connecting a set of points in the hyperbolic plane; (3) Melzak's conjecture which asks what polyhedron with unit volume in Euclidean 3-space has the shortest edgelength (for example, a cube with volume 1 has total edge length 12; there is a polyhedron that has a shorter edgelength--can you figure it out?).

Some nice internet sites are:
Soap bubbles and isoperimetric problems (http://math.berkeley.edu/~hutching/pub/bubbles.html)

Exploratorium soap bubble page (http://www.exploratorium.edu/ronh/bubbles/bubbles.html)

Some open problems in soap bubble geometry (http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf)

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Minimal Surfaces

At each point p on a surface M in Euclidean 3-space, we can compute a normal vector n. Any plane that contains n will intersect the surface in a curve c. For each curve c, we can compute its curvature. As we rotate the plane through the normal n, we will get a set of curves on the surface each of which has a value for its curvature. Let k1 and k2 be the maximum and minimum curvature values at p, respectively. The mean curvature of M at p is H=(k1+k2)/2. Then M is a minimal surface if the mean curvature equals zero at every point. The figures on this page are examples of minimal surfaces. We can use ideas from complex analysis to investigate minimal surfaces. In particular: (1) we can use the Schwarz-Christoffel formula from complex analysis to derive analytic functions that map onto convex polygonal regions. We can then shear these functions and derive the corresponding minimal surfaces and see how they are related to Jenkins-Serrin minimal surfaces which project to convex polygons; (2) we can shear elliptic integrals of the first kind to get a family of minimal surfaces that range from Scherk's doubly-periodic to the helicoid. What families of surfaces do we get, when we shear elliptic integrals of the second and third kind?

Some nice internet sites are:
M. Weber's "Minimal surfaces--introduction"
H. Karcher and K. Polthier's "Touching Soap Films: an introduction to minimal surfaces"
Tech. University Berlin's java applet that graphs various minimal surfaces

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