Properties of Soap Films Energy is proportional to area. Films try to shrink, to minimize energy. Surface tension is constant, for any extent, as long as there are enough soap molecules. Soap films are thin, about 1/100 the thickness of a sheet of paper for ordinary films, and 1/100 of that for the thinnest black films.
Complicated Soap Films Soap films can get much more complicated than simple smooth surfaces, which makes them interesting and more mathematically difficult. What kinds of things can happen where soap films join?
Junctions of Soap Films The equal-angle Y is the only possible way for soap films to meet along a line or curve.
Junctions of Triple Lines Multiple triple lines can only meet four at a time, at equal angles, at a tetrahedral junction.
Films on a Cube Frame A film on a cubical frame with 12 flat surfaces meeting at the center would satisfy the triple-line criterion, but the center point is not tetrahedral, so it would pop into the central rounded square on the right, with four tetrahedral points.
Films on a Trefoil Knot Several films can form on a trefoil knot wire, including one that does not touch the entire boundary.
A Conjecture My Ph.D. adviser, Fred Almgren, conjectured in 1976 in Scientific American magazine that a film could exist on an unclosed wire only if the wire had thickness.
Soap Film Questions What is a good mathematical model for soap films? How can one prove the existence of a soap film on a given boundary? How can one prove a given soap film has absolute minimum area among all competitors?
Classic model of soap films A soap film is an oriented manifold spanning an oriented closed curve. But that cant handle triple junctions.
Duality – the minimum area of a soap film spanning a wire loop is the same as the maximum flux of a incompressible fluid with maximum velocity 1. The flow vectorfield is said to calibrate the surface.
Paired calibration – The minimum film area is equal to the maximum total flux of one incompressible fluid flow for each region, with the velocity difference of each pair of flows at most magnitude 1 at each point.
Proof that a general soap film has minimum area
Hypercube cones are minimal. I was able to show using paired calibration that in space dimension 4 and higher, the cone on a hypercube frame is the minimal area soap film.
Covering Space model of soap films The ambient domain is a covering space of the complement of the boundary, with the boundary wires being branch curves. One sheet is designated the home sheet. A soap film candidate is an oriented manifold separating the home sheet from the others, with the projected sum of 0. (double-layer film) The minimum area of a soap film is the maximum flux from the home sheet into the others of an incompressible flow whose difference between sheets has magnitude at most 1.
Wormhole If the two cuts are in the same universe, we have a wormhole or stargate or portal.
Wormhole Using the other sides of the wormhole, the traveler can move continuously straight forward, but still be in a loop.
Wormhole to Time Machine Einsteins Theory of Relativity says time slows for a moving object, so moving one end of the wormhole slows its clock relative to the other end. But there is no relative motion through the wormhole, so the two clocks agree for a traveler through the wormhole.
Through the Time Machine As the traveler moves back and forth through the wormhole, his personal clock always agrees with the wormhole clock, so to an outside observer he sometimes appears to be in two places at once, and sometimes nowhere.
Paradox A traveler can exit one side and enter the other at exactly the same gate-time, so the traveler is in a closed time-like loop, and his existence has no past cause.
Stargate Construction Needs negative energy cosmic string. Density must be about an Earth mass per inch. No way to hold it, so it must be left to oscillate like a big rubber band in space. It can be made to oscillate with a twist so that it avoids itself at its tightest point, so it can oscillate indefinitely. No gravitational forces to rip you apart.
References K. Brakke, Minimal cones on hypercubes, J. Geom. Anal. 1 (1991) 329-338. http://www.susqu.edu/brakke/aux/downloads/papers/hyper.pdf http://www.susqu.edu/brakke/aux/downloads/papers/hyper.pdf K. Brakke, Soap films and covering spaces, J. Geom. Anal. 5 (1995) 445-514. http://www.susqu.edu/brakke/aux/downloads/papers/covering.pdf http://www.susqu.edu/brakke/aux/downloads/papers/covering.pdf K. Brakke, Numerical solution of soap film dual problems, Exp. Math. 4 (1995) 269-287. http://www.susqu.edu/brakke/aux/downloads/papers/soapdual.pdf http://www.susqu.edu/brakke/aux/downloads/papers/soapdual.pdf Polycut program - http://www.susqu.edu/brakke/polycut/polycut.htm http://www.susqu.edu/brakke/polycut/polycut.htm