How Does This Look Like At The Boundary?

A common recommendation to the layperson who is stranded among a group of mathematicians and doesn’t know what to say is to ask the question above. It will almost always trigger a lengthy and incomprehensible response.

 

Single

 

For example, let’s look at the surface below. It constitutes a building block that can be translated around to make larger pieces of the surface. That this works has to do with the small and large horizontal squares. It is similar to Alan Schoen’s Figure 8 surface, but a bit simpler (it only has genus 4)

 

Pyramid

This surface belongs to a 5-dimensional family about which little is known. The only simple thing I can do with it is to move the squares closer or farther apart. So, how does this look at the boundary? On one hand, when the squares get close, we see little Costa surfaces emerging, as one might expect:

Nearcosta

At the other end of infinity, things look complicated, but depending what we focus on, there is a doubly periodic Scherk surface or a doubly periodic Karcher-Scherk surface:

Nearscherk

Below are, for the sake of their beauty, the two translation structures associated to two of the Weierstrass 1-forms defining this surface. Next week we will study a close cousin of this surface.

 

Flat

 

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Fake Diamonds

Below is something rare. You see two minimal surfaces in an (invisible) box that share many properties, but also couldn’t be more different.Dd4

Let’s first talk about what they have in common: They share lines at the top and bottom of the box, and they meet the vertical faces of the same box orthogonally. This means you can extend both surfaces indefinitely by translating the boxed surfaces around, in which they become triply periodic surface of genus 3.Dd1

How are the different? The red one is a little bit more symmetric and belongs to a 2-dimensional deformation family of the Diamond surface that has been known for about 150 years. You can see how these surfaces deform in an earlier post.

Dd2

The other one belongs to a different deformation family that is only a few weeks old, discovered by Hao Chen, and of which you can see here some wide angle pictures, with clearly different behavior.

Dd3

These surfaces existed right under our nose, but nobody expected them to exist, because minimal surfaces are usually content with a single symmetric solution. Chances are that these surface hold the key in understanding the entire 5-dimensional space of all triply periodic minimal surfaces of genus 3. 

Dd5

2-3-6

I like it when apparently simple things evolve all by themselves into complex objects. Like watching cactus seeds grow into cacti. That was a distraction, but I do like it. Below is a left over piece of mathematics that would have fit nicely into a paper I wrote with Shoichi about triply periodic minimal surfaces.

M236

It is, evidently, quite complicated. To unravel it, here is a smaller portion of it, its seed, so to speek.

Prism

That is a minimal surface inside a prism over a 2-3-6 triangle (which has a right angle, a 30 degree angle, and a 60 degree angle).
The curves in the vertical faces of the prism are symmetry curves of the surface, and reflecting at these faces of the prism extends the surface. The two curves in the bottom and top face of the prism are not symmetry curves, but when you place two prisms on top of each other (by translation), the curves will fit. The pattern the curves make on the prism determines the surface almost completely, there is just one degree of freedom. Here is another, equally pretty, version, using a different parameter.

M236b

Another way to seed these surfaces is through conformal geometry. Below is the conformal image of a circular annulus onto a polygonal annulus bounded by two nested 2-3-6 triangles. The parameter lines are images of radii and concentric circles, respectively. This map is the main ingredient in the Weierstrass representation of all these surfaces. Simple, isn’t it?

Tri