One thing topologists love to do with surfaces is slice them up. You can learn all kinds of things. For example, you can find the Euler characteristic of the surface (if it's closed and continuous), by slicing horizontally from top to bottom and counting the critical points you encounter as you go, adding the number of maxima and minima, and subtracting the number of saddle points. In the case of a sphere, you have one maximum (the north pole), one minimum (the south pole), and no saddles. Sure enough, the Euler characteristic of a sphere is 2.
Now picture a torus standing on end (as if it were a tire swing). When you slice it up from top to bottom, you encounter first a maximum, then a saddle (as you reach the bottom edge of the upper arch of the torus), then another saddle (at the top of the lower, inverted arch), and then finally a minimum, yielding an Euler characteristic of 0.
Students of multivariable calculus know, though, that there are other kinds of critical points besides maxima, minima, and saddle points. There are in fact an infinite number of saddle variants which can be described (very loosely) by the number of upward-curving parts and the number of downward-curving parts. A typical saddle has two upward-curving parts (in front of and behind where you sit), and two downward-curving parts (where your legs go.) A saddle which has three of each we call a monkey saddle-- the extra bend being for the tail.
It turns out that a monkey saddle is a precarious structure: if you tilt it slightly to one side or the other, it falls apart (i.e. it stops being a critical point) and is replaced by two critical points-- each of them an ordinary saddle. To a knife-wielding topologist, this means that a monkey saddle and a pair of regular saddles are basically the same thing.
What does this have to do with these weird pictures? Let's go back to the tire swing. Could we somehow deform the surface so that instead of two saddles in the middle, it had one monkey saddle? It'd be pretty hard, even if you were sneaky and let the surface pass through itself. But what you can do is create such a torus from scratch. And that's exactly what this is: a torus which has one maximum (in figure I), one minimum (figure VII), and one monkey saddle (at the center of figure IV).
Full credit should go to Tom Banchoff, who pointed out this idea in the first place, and drew the two crucial cross-sections (the top edges of figures IV and V) between which the monkey saddle occurs! Also, the drawing style I use borrows directly from techniques George K. Francis uses in his Topological Picturebook.
These drawings are also used to illustrate one of Prof. Banchoff's research articles, Height Functions on Surfaces with Three Critical Points.
A couple of years after finishing these drawings I was approached by a friend of mine who had been in the same class. She was a grad student in mathematics at the time, and wanted something to remember the experience by. The torus immersion, she said, was just the thing. We talked about it over some coffee, and I went home and did some sketches. I had to make the illustration more concise, so that it would look good as one of these.