## Double Periodicity

Last week I explained a really complicated way to get from Scherk’s doubly periodic minimal surface to the helicoid, through a family of Schwarz Diamond surfaces. As was known already to Scherk, this can be done much easier, namely by “shearing” the standard Scherk surface above. I put apostrophes because a simple Euclidean shearing isn’t enough to keep the surface minimal.

Bill Meeks and Hippolyte Lazard-Holly have shown that these are the only embedded doubly periodic minimal surfaces of genus 0 (after taking the quotient by their translational periods). Things get tricky for larger genus.

First of all one needs to distinguish whether the parallel half planes “on top” are parallel to the ones “at the bottom” or not. Today we stick with the case that they are not parallel, and are in fact orthogonal. Then there is just one such surface of genus 1 (I am pretty sure, but I think nobody has written a proof). This was first constructed by Hermann Karcher. It’s pretty clear (and provable) that one can continue like this, creating doubly periodic surfaces with more handles, like the genus 2 example below.

It would be a nice theorem if they all would be unique. But I don’t think so. Below is a picture of a genus 3 surface where the handles are arranged differently.

Proving that this really exists won’t be easy, but interesting, because it would allow one to speculate what will happen if one can shear this surface like the original Scherk surface.

## Once Upon a Time…

This post is about a dying species, the physical book. While book sales are stagnating at best, the number of published books is exploding, meaning that fewer copies of each book are printed. How many of these will survive say a century?

The lucky ones end up in the stacks of large university libraries like the Wells Library shown here. It’s like with animals: In the wild, their life expectancy is much lower than in a zoo.

I grew up in five minute walking distance to a local, privately owned book store. In 15 minute biking distance were two public libraries. These days, privately owned book stores are nearly extinct, and the chains that helped kill them are struggling. Easily obtainable books in print are either classics or bestsellers.

If you want that one exotic book that a friend recommended, you probably need a large library (and be able to read in another language). But they are struggling, too: Space is precious, demand low.

So the aisles are emptied, the books extradited to special auxiliary library facilities, from where you can request them. The happy hours of browsing are gone.

## This Is Not a Helicoid

But almost. It has a vertical axis, lots of horizontal lines, and it twists.

But it is part of something bigger, a triply periodic minimal surface. 32 copies of the above piece, replicated by rotations and reflections, look like this:

This surface sits in a rectangular box over a square. If you identify top and bottom edge of the original squarical helicoid, you get a doubly twisted annulus, which is intimately (confomally, that is) related to a hollow spiderweb:

When squeezing the height down, our non-helicoids become even more helicoidal. When pulling the height up, the helicoids disappear. What we have here is a deformation of the Diamond surface of Hermann Amandus Schwarz.

When he sees this, he will probably just nod.

What happens when we pull a little further? We see doubly periodic Scherk surfaces emerging, stacked on top of each other.

## Gyokuro Omelet

One of most spectacular green teas from Japan is the Gyokuro (jade dew), grown in the shade.

It needs to steep for 1-2 minutes in low temperature (at most 50ºC).

The cup is pale yellow and tastes a little like sea weed.

The steeped leaves are very soft.

Instead of throwing them away, one can use them as a spread. I suggest a simple omelet.

Here is the recipe haiku:

Prepare Gyokuro. Save the leaves.

Beat eggs with Ponzu sauce until smooth.

## Line Congruences (Constant Curvature II)

That revolving a simple, mechanically generated curve, the tractrix, about an axis generates the pseudosphere, a surface of constant negative curvature, seems like one of these unavoidable accidents.

The tractrix has the feature that the endpoints of its unit tangent vectors lie on a line. Thus  dragging one endpoint of a rod of length 1 will have the other endpoint trace a tractrix. More generally, whenever you have a 1-parameter family of lines in the plane, they are typically tangent to a single special curve, the caustic of the line family. Below is the caustic of the normal lines to a parabola.

In space, things get tricky. A 2-dimensional family of lines in space is called a line congruence. They also have caustics or focal sets, i.e. surfaces that are tangent to all the lines, but finding them involves a quadratic equation, so we can expect two of them. Below are the two focal sets for a hyperbolic paraboloid. It is pretty clear that line congruences are hard to visualize.

Note that some of the lines are tangent to the focal sets at some point but intersect it transversally at other points.

By rotating the lines generating the tractix, we obtain a  line congruence whose two focal sets are the pseudosphere and its rotational axis. Note that the segments of the line congruence between the two focal sets have length 1. More generally, a line congruence is called pseudospherical if the segments between the focal sets have constant length, and the surface normals at corresponding points make a constant angle. Remarkably, the focal surfaces of a pseudospherical line congruence are pseudospherical, i.e. have constant negative curvature. Even better, one can start with any pseudospherical surface, pick a point and a tangent vector at that point, and extend this vector to a pseudospherical line congruence whose first focal set is the surface one starts with. This provides a recipe to produce (essentially algebraically) new pseudospherical surfaces.

The bathtub up above is Theodor Kuehn’s pseudospherical surface from 1884. It can be obtained by a line congruence from the standard pseudosphere. Below you see a portion of the pseudosphere with asymptotic lines, and hiding behind, the corresponding portion of Kuehn’s surface.

The last image shows just the lines of the line congruence that has these two surfaces as focal sets.

Did I say this was hard to visualize?

## Patterns of Ice

After almost two weeks of deep freeze, the ice at the local creeks is making feeble attempts to melt.

This has resulted in patterns that are, of course completely useless.

They don’t reduce unemployment, make people smarter, or cure insanity.

But they don’t cause damage, and that is already something these days.

Unbelievable that all this is just water.

## Revolution (Constant Curvature I)

One of the standard elementary surfaces is the Pseudosphere, a surface of revolution of constant negative curvature.

It can be parametrized using elementary function, and the profile curve is the so-called tractrix. Another elementary surface of constant negative curvature is Dini’s surface, where the tractrix is used to produce a helicoidal surface.

From here on, things get tricky. Other such surfaces of revolution require elliptic integrals. Here is the entire zoo (more or less):

Common to all examples is that they necessarily produce singularities. More precisely, there is no complete surface of constant negative curvature in Euclidean space. This is a famous theorem of Hilbert. At the core of the proofs I know is the behavior of the asymptotic lines

Above is the pseudosphere with one family of these asymptotic lines, drawn as ribbons. At the equator, they become horizontal. As the second family is the mirror image of this family, at the equator their tangent vectors become linearly dependent. This shows that while the asymptotic curves exist in the northern and southern hemipseudospheres, the surface itself is singular at the equator, because, alas, on negatively curved surfaces the asymptotic directions are linearly dependent. For the general surfaces of revolution, the asymptotic lines touch both singular latitudes. The image above looks odd because our brain wants to believe that curves on a surface meet at right angles. They don’t.

One of the key features of the asymptotic lines is that they form a Chebyshev net: Opposite edges of the net quadrilaterals have the same length. Thus you can stretch a loosely knitted square mesh over this surface to keep it warm. The standard proof of Hilbert’s theorem continues to show that any net parallelogram has area bounded above by some constant. However, a simply connected complete surface of constant negative curvature has necessarily infinite area, which leads to a contradiction. This was one of the earliest global results in differential geometry.

## Winter

After a mild frost transformed the ground at De Pauw Nature Park into still lives, recent snow fall and deep frost has changed all of that again.

The walls of the former quarry are adorned with icicles, and the ground is a uniform white with occasional bits of vegetation sticking out,

creating patterns of light and dark.

Usually, the little lake is teeming with birds. Now only spare footprints tell me that I am not alone.

It is cold.

## Proof by Example (Push & Pop I)

Here is Push & Pop, a puzzle with a very simple mechanics. It is played on a single strip with a fixed number of fields, occupied by tokens that are stacked on top of each other (as in checkers)

A move consists of taking some of the top tokens of a tower, and moving them onto a field that is as many steps away as you are taking tokens, within the limits of the game board. If the target field is occupied, just place your tokens on top. Note that you can take only pieces from the top, and are not allowed to change their order. In the position above, the possible moves are indicated by the arrows, and you can see the possible new positions below.

You will gladly notice that the color of the tokens does not matter at all at this point. You will also notice that moves are reversible, because undoing a move is also a legal move. Games are in so many ways better than reality.

A typical puzzle using this mechanic is given by two position, and the task is to transform the first into the second using only legal moves. Here is a simple example, played on a board of size three with two tokens of different color, placed on top of each other at the leftmost field. The task is to swap the position of the two tokens:

This is not possible on a board of size 2 (why?), and requires five moves on a board of size 3 like so:

Why should we care? Particular examples can often be used to gain universal insights. In this case, for instance, we have just essentially proven that any puzzle on a board of size at least three can be solved. How so?

We have shown that two adjacent tokens in a single tower can be swapped, if there are two fields available to the right. It does not matter if these fields are occupied or not, because we can just play on top of any existing pieces. It does also not matter if there are tokens below the two we want to swap, and not even if there are tokens on top, because we can move them temporarily out of the way (remembering that moves are reversible).

As the group of all permutations is generated by transpositions (use bubble sort), we can in fact permute the tokens in any single tower to our liking. Finally, to solve an arbitrary puzzle, we first move all tokens (piece by piece, if needed) onto a single tower on the leftmost field, then permute them into the order we need, and then move them into the desired target position.

Here is a puzzle on a board of size 5 with four tokens in 3 colors. You know now that there is a solution. But what is the shortest solution?

To be continued…