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.
For 3D mouse-spinnable images, go here.
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.)
|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.|
|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.|
|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.|
|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.|