k-Noids – The Inner Frame

The Catenoid With a Handle (The Pleasures of Failure III)

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. TwoCatenoids 3

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? Cathandle

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. 3 noid

This works for any order and gets a little boring after a while. 7 noid

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

4 noid 1

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.

2 3 Noid 1

Balance (k-Noids II)

The Catenoid is one of the prototypical minimal surfaces, a building block for more complicated objects. The two openings (ends we call them) spread out to fill almost half of Euclidean space. If we want to have more such ends, we have to chop them off early enough.

Knoid 5 b

This, for instance, is a 5-Noid, because it has five such catenoidal ends. They are quite symmetrically placed, which is not necessary, at lest not to this extent.

4Noid Big

Here is a 4-Noid. The two little catenoids poking out (like eyes??) at the front push their bigger brother and sister backwards, suggesting a rule of balance that must be followed. This is indeed the case: the direction vectors of the ends (the way they poke), scaled to take their size into account, must be in balance. This is one of the many reasons why minimal surfaces are so esthetically pleasing: They keep a sense of equilibrium.

This is convenient for the mathematician, who knows that whatever minimal surface we discover, it will be pretty, but disappointing for the artist, who can’t claim credit for its pre-established harmony.

The images on this page were rendered with Bryce3D. In my first experiments with Bryce3D, I was captivated by the possibility to put alien looking abstract mathematical sculptures into more or less realistic landscapes.

Noidnew

However, while real landscapes have automatically meaning for us just because they exist, it is much harder for imaginary landscapes to acquire an equivalent meaning (maybe with the exceptions of the landscapes we dream about). So I abandoned the capabilities of Bryce3D as a landscape renderer but instead started to explore its immensely complex texture editor. The last image of today is an attempt of a reconstruction. I have lost the Bryce3D scene file, and only a very small version of the rendered scene has survived. So here is the new version, rendered using an old Mac laptop that still can run OS 9 and my old version of Bryce3D.

Catcube

Minimal Surfaces in the Wild (k-Noids I)

Making photorealistic images of minimal surfaces is one thing, but making real models of minimal surfaces and putting them into the landscape is quite something else. In July 2015 I was contacted by the Swiss artist Shireen Caroline von Schulthess who planned to do exactly that. She needed 3D models in order to build large wire frames that would then be wrapped in thin, colorful fabric. These sculptures would serve as loud speakers for recorded voices from local interviews with the topic “wishes”, to be played as an installation at the Lenzerheide Zauberwald festival.

Shireen 1b

Here is the wireframe model of the Finite Riemann minimal surface. Given that I already have difficulties bending a single metal coat hanger into a given shape, I can only admire the skills of Shireen and sculptor Adrian Humbel to accomplish this at this scale.
Below is the partially clothed 4-Noid.

Shireen 2