Topological Stages in the Eversion A generic regular homotopy has topological events occuring only at isolated times, when the combinatorics of the self-intersection changes. These events happen at times when the surface normals at a point of intersection are not linearly independent. That means either when there are two sheets of the surface tangent to each other (and a double curve is created, annihilated, or reconnected), or when three intersecting sheets share a tangent line (creating or destroying a pair of triple points), or when four sheets come together at a quadruple point. The double tangency events come in three flavors. These can be modeled with a rising water level across a fixed landscape. As the water rises, we can observe creation of a new lake, the conversion between a isthmus of land and a channel of water, or the submersion of an island. These correspond to the creation, reconnection, or annihilation of a double-curve, here seen as the shoreline. Figure 8: This minimax sphere eversion is a geometrically optimal way to turn a sphere inside out, minimizing the elastic bending energy needed in the middle of the eversion. Starting from the round sphere (top, moving clockwise), we push the north pole down, then push it through the south pole (upper right) to create the first double-curve of self-intersection. Two sides of the neck then bulge up, and these bulges push through each other (right) to give the second double-curve. The two double-curves approach each other, and when they cross (lower right) pairs of triple points are created. In the halfway-model (bottom) all four triple points merge at the quadruple point, and five isthmus events happen simultaneously. This halfway-model is a symmetric critical point for the Willmore bending energy for surfaces. Its four-fold rotation symmetry interchanges its inside and outside surfaces. Therefore, the second half of the eversion (left) can proceed through exactly the same stages in reverse order, after making the ninety-degree twist. The large central image belongs between the two lowest ones on the right, slightly before the birth of the triple points. In the minimax eversion with two-fold symmetry, seen in Figs. 8 and 9, the first two events create the two double-curves. When these twist around to intersect each other, two pairs of triple points are created. At the halfway stage, six events happen all at the same time. Along the symmetry axis, at one end we have a quadruple point, while at the other end we have a double tangency creating an isthmus event. Finally, at the four "ears", at the inside edge of the large lobes, we have additional isthmus events: two ears open as the other two close. (See [FSK] for more details on these events.) Figure 9: This is the same eversion shown in Fig. 8, but rendered with solid surfaces. Here we start at the top left with a round sphere, and proceed clockwise. We see the creation first of two double-curves (at the top) and then of a pair of triple points (at the top right). (Another pair is created at the same time in back; the eversion always has two-fold rotational symmetry.) Near the bottom right we go through the halfway-model (also seen in the large central image) and interchange the roles of the red-yellow and blue-purple sides of the surface. Continuing around the left side, we see the double-curves disappear one after the other. The three-fold minimax eversion, using the Boy's surface of Fig. 1 as a halfway-model, has too many topological events to describe easily one-by-one, and its three-fold symmetry means that the events are no longer all generic. But we can still follow the basic outline of the eversion from Fig. 10. Figure 10: This three-fold minimax eversion starts (top row, clockwise) with a gastrula stage like that of the two-fold eversion, but the three-fold symmetry means that three fingers reach up from the neck instead of two. They intersect each other (righthand images) and then twist around, while complicated things are happening inside. The lower right and central images are near the halfway model. The two images at the bottom with gaps between the triangles show the double-curves; the Boy's surface halfway-model intersects itself in a "propeller" curve, which then gets replaced by its four-fold cover as we separate the two sheets. Acknowledgments The minimax sphere eversions described here are work done jointly in collaboration with Rob Kusner, Ken Brakke, George Francis, and Stuart Levy, to whom I owe a great debt. I would also like to thank Francis, Robert Grzeszczuk, John Hughes, AK Peters, Nelson Max and Tony Phillips for permission to reproduce figures from earlier sphere eversions. This paper first appeared in the proceedings of two summer 1999 conferences on Mathematics and Art: ISAMA 99 (June, San Sebastián, Spain), and Bridges (July, Kansas). I wish to thank Nat Friedman, Reza Sarhangi, and Carlo Séquin for the invitations to speak at these conferences. I thank Slavik Jablan for formatting this online web version of the paper. My research is supported in part by NSF grant DMS-97-27859. REFERENCES