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. 


Octagon 01

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.

Octagon clp

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:

Clp 0

Let’s begin to rotate through the associate family. For angle π/16, we see how the touching vertices are being separated.

Clp 16

At π/4, we get a nice symmetric piece, but translational copies will intersect so that the surface will not remain embedded.

Clp 4

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.Clp 2



The Lidinoid (H 2)

All minimal surfaces can be locally bent in a 1-parameter family of associate minimal surfaces. In the right context this is just a rotation. The best known example is the deformation of the catenoid into the helicoid. 

H pieceRemarkably, the triply periodic Meeks surfaces, rotated in this sense by 90 degrees, are again in the Meeks family. Very curiously, there are two more known cases where an associate surface of a triply periodic minimal surface is again triply periodic and embedded. One of them is Alan Schoen’s Gyroid, the other is Sven Lidin’s Lidinoid. While the Gyroid occurs in the associate family of the P/D surface, the Lidinoid arises in the associate family of one particular H-surface. To see how this happens, let’s start with a top view of that H surface. When we start rotating in the associate family, the vertices of the three triangles that meet at the center of the image move apart:


But the other vertices then move towards each other, so that, after about 64.2°, they come together:

 Lidinoid small

Not only the vertices fit, but again the surface can be extended by translations in space. Here is a view of a much larger piece.

Lidinoid big

If you are good at cross-eyed viewing, here is a stereo pair of a side view:



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Indiana is mostly flat, so there are no spectacular views from mountain peaks. Every little dropoff becomes an attraction, in particular when flowing water is involved.

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One hopes that heavy rainfall would make things better. Up above you can see a portion of the Upper Cataract Fall, which continues over a few cascades and has a total height of 45 feet. 

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Not spectacular, but quite pleasant. The site claims this is the largest waterfall by volume in Indiana. I guess this depends on when you measure. 

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Equally pretty are the flirting trees on the shore. The words “upper” suggests that there is also a Lower Cataract Fall. It is supposed to drop 30 feet. This is it:

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The second problem with Indiana is that the ground drains poorly. So, after heavy rainfall one gets heavy flooding. In this case, the lake behind the Lower Cataract Fall has risen to the height of the top of that fall, making it disappear. Bad luck?

Schwarz Hexagonal Surface (H-1)

One of the early minimal surfaces I have neglected so far is the H-surface of Hermann Amandus Schwarz.H double

Think about it as the triangular catenoids. Two copies make a translational fundamental domain, i.e. the 10 boundary edges can be identified in pairs by Euclidean translations, thus making the surface triply periodic. As a quotient surface it has genus 3, which implies that the Gauss map has 8 branched points. They occur at the triangle vertices and midpoints of triangle edges. Thus the branched values lie at the north and south pole of the sphere, and at the vertices of two horizontal equilateral triangles in parallel planes. In particular, they are not antipodal, making these surfaces the earliest examples of triply periodic minimal surfaces that lie not in the 5-dimensional Meeks family.

H catenoid

Above is a larger portion of the H-surface with the triangle planes close to each other. In the limit we get parallel planes joined by tiny catenoidal necks. When we pull the planes apart, we get Scherk surfaces:


H Scherk

The spiderweb for this surface looks also pretty:

H spider

Among the crude polyhedral approximations, there is one that tiles the surface with regular hexagons. The valencies are 4 and 6, so the tiling is not platonic.

H poly

Next week we will look at one of its more surprising features.


The Other Side (Trevlac Bluffs Nature Preserve)

The Trevlac Bluffs Nature Preserve is divided in half by Beanblossom Creek, and each part has its own hiking trail. DSC 0789

There is no connection, because one side of the creek consists of a very steep and (for Indiana) pretty tall cliff. Most people will probably prefer the well maintained southern part where you get a decent hike with neat little sights like the shed up above, or the pond below.

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On top, you get a view of the creek, always obstructed, even in winter.

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I found the other side much more exciting. It is mostly wetland, and the trail is rather a maze of paths. 

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You meander around and inhale the intoxicating atmosphere of water at work.

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After heavy rainfall, finding a path to the creek is not trivial, but well worth the effort.

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Just don’t do it on a sunny day.

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Dissections and Area

Whenever need to explain what Mathematics is about, one of my favorite examples is the concept of area. The existence of an elementary notion of area hinges on the fact that any two Euclidean polygons have the same area if and only if they are scissor congruent, meaning that they can be cut into congruent pieces using straight cuts. To see this, it suffices to show that any polygon can be dissected into a square. Rect2square

The example above shows how to dissect a well-proportioned rectangle into a square. Here, well-proportioned means that the rectangle is not more than twice as tall than wide. If a rectangle is not well-proportioned, a few cuts parallel to the edges will make it so. Thus any two rectangles of the same area can be dissected int each other. We will use this later.


Next we show that any polygon can be dissected into triangles. By induction, it suffices to find a secant inside the polygon. To find this secant, pretend that the polygon is actually the floor plan of a room, and we are standing at one vertex V . The two adjacent walls lead to two vertices A and B which we can see. If we can see yet another vertex W from our position, we have found our secant VW. If we can’t see another vertex, nothing obstructs our view in the triangular region formed by A, B and V , and thus A and B can be joined by a secant.

As a further simplification, we cut all triangles into two pieces along one of their heights so that all triangles become right triangles.

Now we have a collection of right triangles, which will need to be dissected into a single square.

Triangle2squareTo do so, we dissect each right triangle into a rectangle. This can be done as shown above by dissecting the triangle into two pieces along a segment parallel to one of the legs and dividing the other leg into equal parts.

This leaves us with a collection of rectangles that most likely have different dimensions. So we dissect them into new rectangles that all have all height 1, using the example at the beginning. 

Then, the new rectangles can be lined up edge to edge along their sides of length 1 to form one very wide rectangle that finally can be dissected into a square.


As this was nice and easy, here a challenge: In our dissections, we were allowed to translate and rotate the pieces arbitrarily. What about if we forbid rotations? Can you dissect an equilateral triangle into finitely many pieces and translate them so that the result is the same triangle upside down? Or, can you cut a square, translate the pieces, and thereby achieve a 45 degree rotation of the square? 


First Times (California ’93)

My little series with  pictures from 25 years ago continues with my first hiking trip in California. The idea was to drive up to the trail head of White Mountain Peak, and hike the dirt road to the peak.


We drove through Yosemite at night (which I hadn’t seen before) and camped at my first hot spring in Owen’s Valley. Soaking in warm water while around you everything freezes and the sky is full of shooting stars convinced me that this had been a good idea. We made it past the Bristlecone Pine Trees, but the car didn’t make it to the trail head (my first car break down).


We didn’t give up though but continued on foot. The landscape up there (above 10,000 feet) is high elevation desert. 


After two hours or so we reached the observatory and the actual trailhead. Hiking appears very easy: You just follow the dirt road.


What is not so easy, however, is the high elevation. Two of use got altitude sickness, including myself (first time!). That was interesting. It started off with gradually worsening headache.


After a while, my vision got blurry, and me and the other victim turned back to the observatory. While we waited for the two others to return from the summit, we chatted with the friendly personnel. By 10pm, the two other hikers had returned, and we were lucky to hitch a ride on a pickup truck back to our car.

The next morning we stopped for my first visit at Mono Lake.