The surface shown in "Triple-Point Twist" is one from a series of surfaces described by David Mond and Washington Marar in "Real map-germs with good perturbations" , where they analyze a number of germs of singularities of surfaces. This particular example is a ruled surface given by the equations

(*x*,*y*,*z*) =
(*u*, *v*^{3} + *cv*,
*uv* + *v*^{5} + *cv*^{3}),

where *c* is a parameter that can be varied. For values of
*c* greater than 0, the surface has no self-intersection, but for
values of *c* less than 0, a triple point and two pinch points
appear. For the image shown in the exhibit, *c* = -1, but an MPEG
movie is available showing a series of different values of *c* as
it varies from -1 to 1. For each fixed value
of *u*, allowing *v* to vary produces a planar curve in the
plane *x* = *u*. The rulings for the surfaces
are the straight lines produced when *v* is held fixed and *u*
is allowed to vary. One of the MPEG movies shows the surface being
swept out by these ruling lines.