Tag: Minimal Surface

Periodic k-Noids (Minimal Surfaces in the Wild II)

The k-Noids that Shireen built last winter will keep roaming the Swiss landscape, from May to August in Wülflingen. Maybe it is time to corral them. My suggestion is to build fences of catenoids. The most classical one looks like this:

Fence1

It is, technically, a translation invariant minimal surface that has two ends and genus 1 in the quotient. A simple generalization and an even simpler 90 degree rotation leeds to towers with catenoidal openings.

Fence2

If that isn’t safe enough, you can have them with double walls (i.e. with genus two in the quotient) like so

Fence3

or so:

Fence4

All these examples have many ends in the quotient. The surprise is that there also is the elusive Uninoid which only has one catenoidal end in the quotient, namely by a 180 degree screw motion:

Fence5

Here things get tricky. Michelle Hackman has found more complicated versions of this in her thesis. Here is a Uninoid that is invariant under a screw motion with quarter turn.

Fence6

Minimal Graphics (Spheres IX)

This post in the Sphere Series is motivated by the recent Circles post. It’s easy enough to conceive a generalization where we place spheres with centers at the points with integer coordinates in space, and set the radius so that something interesting happens.

There is a problem, though. We could visualize the 2-dimensional circle case because we could look onto the plane from our privileged position in 3-space. To do the same with spheres, we would need to step outside 3-space into 4-space. Let’s not do that.

Instead, let’s look at the simplest case of circle intersections. We can think of the quarter arcs as deformed straight edges of squares.

Quarters 01

To make things visible, we have to remove some of them, and a natural choice is to remove every other arc, like so:

Quarters wiggle 01

A similar approach works in three dimensions. Here, the spheres are arranged in a cubical lattice, and we can think of this as tiling by cubes where each cube has been replaced by an inflated sphere.

Cubespherical

This would still be too busy, so I have removed some of the spherical shards. The choice for that is suggested by a minimal surface, the P-surface of Hermann Amandus Schwarz.

600b

You can think of it as consisting of plumbing pieces that have connectors in six directions: up, down, left, right, front, back. There is a coarse polygonal approximation by it, using squares. Both the original minimal surface and its polygonal approximation divide space into two identical parts. A rat could not tell whether it lived on the insid or outside of the plumbing system.

Morton4

If we push the squares a little as to create four-sided pyramids, alternating the direction, we get the prototype of the model of sphere shards. In the spherical version, the shards meet just at the corners, leaving enough space so that light can get through.

Morton3

To make the sculpture more interesting, I have varied the colors, and moved it (sort of) off center. I feel it is a a visual representation of minimal music. Granted, there are many kinds of minimal music, and I do like many of them, but not all. The one I have in mind here would have to be composed by Steve Reich.

Shards3b

This would make a nice pendant sculpture. As material, I would prefer ceramics, not glass.