This surface has been discovered by Scherk in 1835 and has the property that it locally minimizes area. To make this more clear, imagine the surface on the left is rigid and cut out a small patch, then the patch you have has the smallest area with the given border. In other words, if you keep the curve at the border fixed, bending the interior of your patch will only increase its area. Surfaces with this property are called minimal surfaces.
Soap films also have that property! If you dip a wire frame (make it crooked!) into a soapy solution, the soap film you see is the surface that has minimal area among all the surfaces that span the wire.
Take a point on a surface (imagine the surface to be an apple or, more challenging, a pear), there is a direction that comes straight out of the surface at that point, this called the normal direction. We now make cuts of the surface (your apple or pear) with planes that contain the normal direction. Every cut gives a curve, every curve has a curvature, which measures how tight the curve is.Minimal surfaces have mean curvature equal to zero. Roughly speaking, at a given point, all the curvatures average out to zero.
|In the picture to the right, we have two cuts (don't pay attention to the skimming cut of the tangent plane): one that gives a curve upward and one that gives a curve downward. The curvature of these two will have opposite sign (which one is positive and which one is negative is just a matter of convention: you choose). The mean curvature is the average of the largest curvature and the smallest curvature of all the possible cuts. If the mean curvature is zero, you have a minimal surface.
Let's go back to our apple. If you cut an apple, all the curves will go in the same direction so your average is either positive or negative (depending on convention), but definitely not zero. So it looks like in order to be minimal, a surface has to curve one way in some direction, and the opposite way in another direction - like the picture of a saddle here. Not only that, but exaclty the same amount so that the average ends up being zero.
A pear has a little bit of that going for it at its "waist". Near the bottom or the top, the surface just bends one way. Unlike pears, minimal surfaces have the remarkable property of having zero mean curvature at every point.
Finally, here is a plug for the Bloomington's Virtual Minimal Surface Museum. Go to the Gallery and see pictures of exotic minimal surfaces!