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Soap Films on the Borromean Rings

The Borromean Rings are three rings linked so that any pair of rings are not linked, but all three are (i.e. cannot be pulled apart). The Rings support many soap films, and I picture here all of those that I could find. If you think you have another one, please let me know. There are other films that might exist on Rings of different shape; for example, a film with a quadrilateral center (much like in an octahedral film) is unstable here, but I can get it physically using a rectangular set of Borromean Rings and soap solution. Note in general the similarity of the full films to the octahedral frame films.

The images were made with my Surface Evolver program. The rings were modeled as zero-thickness curves. The thick rings you see were added afterwards for display purposes.

Note that the films obey Plateau's Laws of Soapfilms: three films meet along a curve at equal angles, and three such triple curves meet at a "tetrahedral point" at equal angles. Further note that where triple curves meet rings, the curves come in tangent to the rings, and there is actually a short gap along the ring between where two triple curves meet it, although that gap is too small to be apparent here. Also note (in the bottom row) that it is possible to have a ring partially bare.

And for better 3D mouse-spinnable images using the upcoming WebGL 3D technology, go here. (Only works on Firefox 4 Beta, suitably configured. Before loading, browse "about:config" and set "webgl.enabled_for_all_sites" to "true"; this setting will be permanently remembered.)

bare rings orientable seifert manifold unorientable manifold unstable manifold
1. Bare Borromean rings. 2. Stable orientable manifold (no triple lines) spanning the rings. This is known as the "Seifert surface" in knot theory. 3. Stable unorientable manifold. 4. Unstable manifold resulting from poking out all six small triangular areas in the hexagon-center film. This film is unorientable.
hex center pentagon center square center tetra center
5. Soap film spanning the rings with a hexagonal center. 6. Soap film spanning the rings with a pentagonal center. 7. Soap film spanning the rings with a square center. 8. Soap film spanning the rings with a tetrahedral point center. This film has the least area among the four fully spanning films.
tetra poke one hex poke hybrid hex-poke-2adj
9. Film resulting from poking out one of the inner triangles from the tetrahedral-point-center full film. 10. Film resulting from poking out the central hexagon in the hexagon-center film. 11. Film resulting from poking out two opposite triangular areas in the hexagon-center film. This film can be viewed as the union of an elliptical film on one ring with a twisted strip film on the other two rings, with the intersection between them resolved by splitting the quadruple lines into two triple lines. 12. Film resulting from poking out two adjacent small triangular areas in the hexagon-center film.
bare lobe bare lobe poke small bare lobe poke large tetra bare lobe
13. Film resulting from poking out one outer lobe in the hexagon-center film. This film exists even with zero thickness rings. 14. Film resulting from poking out one outer lobe and opposite small triangular face in the hexagon-center film. 15. Film resulting from poking out one outer lobe and one of the large triangular faces in the hexagon-center film. 16. Film resulting from poking out one outer lobe of the tetrahedral-point-center film.
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