In the Beginning, Leonhard Euler created the Calculus of Variations and gave many examples. In one of them he proved that if you want to minimize the area of a surface of revolution (among surfaces of revolution), you will get a piece of a catenoid or plane.

The story is not as simple as it seems, because catenoids stop being minimal at a certain size. Above, for instance, two catenoids that have the same two circles as boundaries. Clearly only one of them has minimal area.

One of the early stories of failure was the attempt to add a handle to the catenoid. Maybe one could even save area this way?

Not so, as the handle doesn’t even extend far enough to close up. Rick Schoen proved more generally that catenoids don’t come with handles of any sort and number. But one can try other things, like making the catenoid more symmetric. This is of course silly, but Luquesio Jorge and Bill Meeks did just that by turning the reflectional symmetry at a horizontal plane into a dihedral symmetry, thus creating the k-Noids.

This works for any order and gets a little boring after a while.

But, somewhat surprisingly, one can add a handle to these k-Noids when k is at least 3:

Like last week, k does not need to be an integer, and one can see clearly what goes wrong when one pushes k below 3: Here we have a broken catenoid with a handle.