# 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.

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.

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.

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.