## From H to P and beyond (H-3)

In March we had a look at the Schwarz H surface, and it is time to revisit it. We begin by turning it on its side, for comfort: Then horizontal lines and vertical symmetry planes cut the surface still into simply connected pieces like this one The H-surfaces form a natural 1-parameter family with hexagonal symmetry. It turns out that in this representation one gains another parameter at the cost of  losing the hexagonal symmetry. This allows to deform the H-surface into new minimal surfaces, and the question arises what these look like. To get used to this view, below is yet another version of the classical H-surfaces near one of its two limits. The new deformation allows to shift the catenoidal necks up and down, until they line up like so: This surface is a member of the so-called orthorhombic deformation of the P-surface of Schwarz so that we can deform any H-surface into the P-surface, and from there into any other member of the 5-dimensional Meeks family.

This is remarkable because the H-surface does not belong to the Meeks family, but to another 5-dimensional family of triply periodic minimal surfaces that is much less understood. The final image is another extreme case of the newly deformed H-surfaces: ## Boundary Considerations, Part II

As promised, today we will look at a close cousin of last week’s surface. A good starting point is the CLP surface of Hermann Amandus Schwarz, about which I have written before. Up above are four copies of a translational fundamental piece. There are horizontal straight lines meeting in a square pattern, vertical symmetry planes intersecting the squares diagonally, vertical lines through edge midpoints of the squares and horizontal symmetry planes half way between squares at different heights. What more could one want? Well, CLP has genus 3, and we wouldn’t mind another handle. There are various ways of doing that, and one of them leads to today’s surface, shown above. For adding a handle we had to sacrifice the vertical straight lines, but all other symmetries are retained. These are, in fact,  essentially the same symmetries we had in last week’s surface, except that there, the squares in consecutive layers were shifted against each other. The similarities go further. Again we can ask how things look at the boundary. Pushing the one free parameter the the other limit, gives us again doubly  periodic Scherk surfaces and Karcher-Scherk surfaces. There is a subtle difference (called a Dehn twist), however, how the two types of Scherk surfaces are attached to each other in both cases. Finally, as usual, the cryptic rainbow polygons that encode everything. Today, the two fit together along their fractured edges, which has to do with the period condition these surfaces have to satisfy.

## 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. 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. 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. 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. 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. ## The Octagon (CLP-1)

Let’s start with an equation: y²=x⁸-1. Solving for y is easy, because for each x we appear to have just two choices for the sign, good and evil. If we do this in the complex plane, the set of solutions therefore looks like two copies of the x-plane. There is a little problem at the eighth roots of unity, because there, good and evil coalesce. A good way to imagine this is to think about the (extended) complex plane as two disks, and of each disk as a regular octagon, with vertices at these eighth roots of unity. Then it takes four such octagons to build the solution space of the equation y²=x⁸-1, and we need to have four octagons at each vertex coming together, alternating between good and evil. Luckily, this can be done in the hyperbolic plane, using a tiling by regular right-angled hexagons.To get an idea how these are glued together, it helps to think about the equation in the form x⁸=y²+1. This represents the same solution space as 16 copies of a single triangle, with vertices at the octagon centers as shown above. Thus the entire solution space can also be obtained by gluing together the edges of the 16-gon above, where the identifications are indicated by the (extended) edges of the central octagon.

Wouldn’t it be nice if we could visualize this in ℝ³? This is indeed possible if we are willing to conformally bend our octagon a little so that every other edge becomes a straight segment, and the other edges lie in planes that meet the octagon orthogonally along that edge. This allows to extend the octagon by rotating and reflecting about its edges like above, which shows four such hexagons, i.e. the entire solution space. If you do this right, you get one of the many views of the CLP surface of Hermann Amandus Schwarz. CLP stands for crossed layers of parallels. This is once again a triply periodic minimal surface. Here is another translational fundamental piece that corresponds to the 16-gon: Let’s begin to rotate through the associate family. For angle π/16, we see how the touching vertices are being separated. At π/4, we get a nice symmetric piece, but translational copies will intersect so that the surface will not remain embedded. At π/2 we meet the conjugate surface of the CLP surface. The amusing point here is that it is congruent to the CLP surface, a feature it shares with the Enneper surface and one surface in the family of Riemann’s minimal surfaces. ## 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. ## Safely Footed Spiderwebs

… et je continue encore de fouler le parvis sacré de votre temple solennel… Let’s talk again a little about triangles. The last time I wrote about triangles is not quite a year ago, and it didn’t help. What you see in this post are all triangle that have their vertices at the same place, the third roots of unity, to be precise. They are, however, not Euclidean triangles with straight edges, but curved ones, with circular edges. The first three are equiangular still, making angles of 10, 90 and 240 degrees at the corners, respectively. The spiderwebs are conformal images of polar coordinates on the disk, thus illustrating the Schwarz-Christoffel formula for circular polygons. The bat down below is a neat optical illusion, too: Would you think that the vertices are at the corners of an equilateral triangle? The theory behind this is based on Schwarzian derivatives and the Schwarz reflection principle, so clearly Hermann Amandus Schwarz owns all this.
It is also intimately connected to hypergeometric functions and much more recent mathematics. And there is some mystery, still. While circular triangles are safe (they are determined by their angles, up to a Möbius transformation, and the Schwarz-Christoffel formula will deliver), quadrilaterals are not. Even Euclidean straight & right quadrilaterals can be differently shaped rectangles, and things get worse with circular ones. In this case, the Schwarz-Christoffel formula will have some extra parameters, the so-called accessory parameters. Changing the will not change the conformal nature of the quadrilateral nor its angles, but its “bulge”. More about this later.

## Golem (Fattend Skeletons)

Today I want to look at decorations of simplicial graphs. As an example, here is a decoration of last week‘s skeleton: The vertices of the two skeletons have been replaced by tetrahedra, oriented and scaled so that vertices of neighbors touch. This should explain what I mean by a decoration: A geometric construction that consistently modifies the graph, in order to obtain something with similar symmetries (many) and possibly other desired properties. Another simple and well known decoration is that by Voronoi cells: We replace each vertex by the set of all points that are closer to that vertex than to any other vertex. In this case, the Voronoi cells are truncated octahedra, as shown in another post. Instead of replicating it here, one can also pass to the dual tiling, which is by rhombic dodecahedra. Try viewing it cross eyed. This is another polyhedral version of the Diamond surface of Schwarz. Like the one obtained from the truncated dodecahedra, the polyhedral approximation shares the conformal structure (by sheer symmetry).

There is another decoration that is quite remarkable: Here, we have replaced each vertex and each edge of the original graph by an octahedron, properly scaled an oriented. We thus get two fattened skeletons that are congruent and disjoint. All faces are equilateral triangles, and all vertices have valency 6. There even is enough room between them to fit in the truncated octahedra, as one can see in the last image. This image also shows how to position each octahedron within the cubical lattice: The central octahedron has its vertices along the edges of the lattice, dividing each edge in the ratio 1:3. That ratio ensures that the other three octahedra in the image become in fact regular. More about this next week.

## Walls and Connections

The cubical lattice is a seemingly simple way to arrange spheres in space. By connecting spheres that are closest to each other, we get a line configuration I have also written about before. Let’s increase the complexity by adding another copy of the same configuration, shifted by 1/2 of a unit step in all coordinate directions. This is sometimes called the body-centered cubical Bravais lattice. We can also recognize here the two skeletal graphs of the two components of the complement of the Schwarz P minimal surface. This means that the P surface will separate the yellow and the red lattices.

Now we would like to connect the two separate systems of spheres with each other. Note that each yellow sphere is surrounded by 8 red spheres (and vice vera), at the vertices of a cube centered at the yellow sphere. This suggests to connect the yellow center to just four of these red neighbors, by choosing the vertices of a tetrahedron, as to obtain a 4-valent graph. Like so: While this is still simple, it starts to look confusing. The new skeleton has again two components, and again they can be separated by a classical minimal surface, the Diamond surface of Hermann Amandus Schwarz.

All this should remind us of the Laves graphs, which are skeletal graphs of the gyroid. You can see that these skeletal graphs have girth 6. Below is a larger piece of the D-surface. Everything here is triply periodic and very symmetric. In contrast to the Laves graph, these here have no chirality. Next week, we will decorate these skeletons a little.

## Praising the Underrated (Enneper I)

In 1760 Joseph Lagrange writes, after establishing the minimal surface equation of a graph and observing that planar graphs do indeed satisfy his equation, that “la solution générale doit ètre telle, que le périmètre de la surface puisse ètre détermine a volonté” — the general solution ought to be such that the perimeter of the surface can be prescribed arbitrarily.

For a hundred years, little progress was made to support Lagrange’s optimism. Few examples of minimal surfaces were found, and most of them with considerable effort. Then it the second half of the 18th century, it took the combined efforts of Pierre Ossian Bonnet, Karl Theodor Wilhelm Weierstraß, Alfred Enneper, and Hermann Amandus Schwarz to unravel a connection between complex analysis and minimal surfaces that would become the Weierstrass representation and revolutionize the theory. One piece in this story is Enneper’s minimal surface. Enneper was not so much after minimal surfaces but after examples of surfaces where all curvature lines are planar. This was immensely popular back then, and the long and technical papers are mostly forgotten. Above is an attempt to visualize the planes that intersect the Enneper surface in its curvature lines. Visually easier to digest are the ruled surfaces that are generated by the surface normals along the curvature lines, because here the ruled surfaces and the Enneper surface meet orthogonally. While not planar, they are still flat, and invite therefore a paper model construction (that one can do for the curvature liens of any surface): Print and cut out the five snakes. The orange centers are the curvature lines. Also cut all segments that go half through a snake, and fold along all segments that go all the way through a snake, by about 90 degrees, always in the same way. Then assemble by sliding the snakes into each other along the cuts, like so: The three long snakes close up in space and need some tape to help them with that. Here is a retraced version of the same model which might help. ## 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. To make things visible, we have to remove some of them, and a natural choice is to remove every other arc, like so: 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. 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. 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. 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. 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. This would make a nice pendant sculpture. As material, I would prefer ceramics, not glass.