Boys Surface (After Bryant-Kusner) Hermann Karcher See the moebius strip first. The non-orientable surfaces are "one-sided", and this concept can best be understood if one starts from a Moebius Strip. The Klein Bottle is easier to visualize than the Boys surface. Each meridian of the Boy surface is the center line of a narrow Moebius band, for example use "Set u,v ranges" to set umin = - 0.998, vmin = 6.1. The "equator" of the Boys Surface is a different Moebius band. It has three half-twists instead of one. The standard morph begins with this Moebius band and widens it until the Boys Surface is complete: aa = 0.5,\ vmin = 0, vmax = 2\pi, umax = 1, 0.9 > umin > 0.002 . The Boys Surface is really a family of surfaces. Boys, in his dissertation under Hilbert, constructed an immersion of the projective plane. Being non-orientable implies that no embedding is possible. Boys Surface has besides its self-intersection curves only one more serious singularity, namely a triple-point. His construction was topological. Apery found algebraically embedded Boys Surfaces. They carry one-parameter families of ellipses. The Bryant-Kusner Boys surfaces are obtained by an inversion from a minimal surface in R^3. The minimal surface is an immersion of S^2 - 6 points such that antipodal points have the same image in R^3. The six punctures are three antipodal pairs, and the minimal surface has so called "planar ends" at these punctures. In this context it is important that the puncture in the inversion of a planar end can be smoothly closed by adding a point. The closing of the three pairs of antipodal ends thus gives a triple point on the surface obtained by inversion. Explicitly: Let M(z) = Re( a(z) V(z) ) + (0,0,1/2) where a(z) = 1/(z^3 - z^{-3} + sqrt(5) ) and V(z) = ( i ( z^2 - z^(-2) ) , z^2 + z^(-2) , (2 i/3) ( z^3 + z^(-3) ) ) Then Boys(z) is obtained by inverting M(z) in the unit sphere: Boys(z) = M(z)/ || M(z) ||^2