Given a fixed boundary in nD, what n-1 dimensional hypersurface with this boundary has the minimum area? This is the question to answer in the field of minimal surfaces, studied by Ken Brakke and John Sullivan of the Geometry Center. In three dimensions, this question becomes: given a boundary in 3D, such as a loop or shape made out of wire, what is the surface with minimum area with this wire as its boundary? The answer should be quite familiar to everyone; the soap film formed by dipping a wire frame in soap solution gives a surface of minimum area with that boundary.
Sometimes these surfaces are unexpected. For example, a tetrahedral frame makes a soap film with planes starting at each edge, all meeting at a point in the center, whereas a cubic frame makes a soap film with surfaces headed towards the center but which meet at an interior square with rounded edges. (Many science museums have soap bubbles and frames for people to play. It is also not difficult to make solution and wire frames.)
Although it is easy to obtain minimal surfaces, they are mathematically difficult to describe. Even for the simple case of a cubic boundary, there is no known mathematical equation which describes the soap film. Even after gaining an understanding of the mathematics, researchers always are going back to soap films. "Whenever I get stuck, I take out the wire frames and soap solution again," says Brakke.
Sullivan looks at n-1 dimensional minimal hypersurfaces in nD, where n is arbitrary. He observes that there are hypersurfaces which are locally minimal but not necessarily globally minimal; in other words, two different surfaces spanning a given boundary may have the property that each minimizes area with respect to perturbations in a small neighborhood of each point of the surface. For example, in 3D, given two parallel circles as a boundary wire, one can get two different soap films: namely, a flat disk inside each circle separately, and a catenoid connecting the two circles. The catenoid surface is the shape you get if you form a surface of rotation using the St. Louis Arch, rotating around a line above it and in its same plane.
Sullivan finds general conditions to determine which of the locally minimal surfaces actually has the minimum area. His conditions apply any arbitrary dimension, so he does not actually have to work specifically with problems in 4D.
Brakke looks at higher dimensional singularities in minimal surfaces. Singularities are just the possible ways in which a surface could be lacking in smoothness. For example, in the tetrahedral boundary case, the planes starting at the edges intersect three at a time along lines toward the center. These lines are singular. Also, the planes all meet at a singular point in the center.
The two singularities above are both mathematical cones on some geometric object, where the cone on an object is defined as all the lines connecting that object to the origin. The first singularity was three planes meeting along a line at 120 degrees. This is the cone on the vertices of a triangle extended into 3D. The second singularity was six planes meeting at a point. This is the cone on a tetrahedral wire frame.
Jean Taylor proved the two types of singularities described are the only possibilities in three dimensions. See the article on Taylor's work in the May/June issue of The Sciences. Thus Brakke is trying to classify singularities for minimal hypersurfaces in higher dimensions.
Since each dimension builds on the singularities of the previous dimensions, Brakke concentrates his work specifically on 4D. He has been successful; using his program Surface Evolver to help guess possible surfaces with singularities, then checking his guesses analytically, Brakke proved that the 3D cone on the 2D hypercube frame in 4D is a minimal surface. In fact, in all dimensions higher than three the n-1 dimensional cone on the n-2 dimensional hypercube frame is a minimal surface with a singularity.
The new singularity, along with singularities obtained by extending 3D surfaces to 4D hypersurfaces, is almost the whole story in the 4D case. Brian White proved that all possible singularities occur on flat sided cones in 4D; the result is specific to 4D and not true in 5D. Based on White's result, Brakke was able to show that there were only a finite number of cases of singularities in 4D minimal surfaces; for all but the simplex frame, hypercube frame, and one other case, he proved that these surfaces were not minimal. Brakke is currently trying to prove or disprove this last case; this would complete the 4D classification.
Brakke says that even after his work in 4D, he is unable to really visualize it. His responded with skepticism to the idea that anyone can visualize 4D. He said, "Give them a test. Show them a bunch of projections of similar 4D objects and ask them which are pictures of the same object. Show them one 'side' of a 4D object, and have them describe the other 'side.' I think you'll find that nobody can really visualize 4D. The best they can do is try to understand a few theorems and see that the objects obeys these theorems."