The catenoid is a geometric
surface which can be generated by rotating a
catenary curve (y =
cosh(x))** symmetrically about the
x-axis. Visually, it looks somewhat like a
spool shape which flares out at the ends,
exponentially.
One significant property of the catenoid is that it is a minimal surface, meaning its mean curvature is exactly zero everywhere (although its gaussian curvature is always strictly negative). To say a surface is "minimal" means, roughly speaking, that at any given point its two principal radii of curvature are equal and opposite; in a small neighborhood of any point, it looks like a saddle surface, curving outward in one direction, and curving inward by the same amount in the perpendicular direction. The catenoid was the first nontrivial minimal surface discovered by mathematicians (nontrivial meaning besides flat surfaces like a plane).
Another way in which this shape has geometric significance is that it is geometrically equivalent (isometric) to the helicoid. The helicoid is roughly a "spiral" surface, vaguely reminicient of a DNA strand. To say that these surfaces are isometric to each other is to say there is a way of deforming one surface without distorting it (without stretching or contracting it) until it becomes the other surface. To see why this might be interesting, I present a method of constructing the catenoid by using a helicoid.
How to construct a catenoid using common household objects:
You will need:
First, make a helicoid using the popsicle sticks and glue. This can be done by pasting one stick on top of another, each one
rotated by a very slight
angle with respect to the one below it. Do this until you complete a full
360 degree turn. You have just constructed (part of) a helicoid.
When the helicoid is completely dry, take the silly putty and smash it up against one side the helicoid, spreading it out to cover the entire surface.
Now carefully remove the silly putty, distorting it as little as possible, and then curl it into a spool shape, sticking the top edge to the bottom edge. It should naturally tend to flare out at the edges. You have now constructed a catenoid out of silly putty. It may not be very stable, but you can check to see that the cross section looks like a catenary curve. This experiment has been done by mathematicians using wax and metallic structures, but this is a more low-budget version.
So, the helicoid can be deformed into the catenoid without distorting it. If we were to imagine a grid placed on the helicoid which we could use to measure distances along the surface, this grid would be completely preserved by the transformation (because the surface was not stretched or contracted anywhere). This means the distance function on the helicoid is the same as the distance function on the catenoid. This is why they are said to be geometrically equivalent, or isometric.
**This is really a special case; a general catenoid is the surface of revolution with profile curve described by y = a cosh(bx), which is not, in general, a minimal surface. The generalizations of this discussion are relatively intuitive, so I omit them for the sake of brevity.