For old-fashioned 2D images and general commentary, see here.
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. |