What to keep…

My first visit to Dresden took place in the early 1990s. It was a foggy day in December, and one of my lasting memories is the enormous pile of rubble in the city center. Dresden331

The ruins of the Frauenkirche hadn’t been touched since the bombing at the end of the Second World War, but after the reunification of Germany, a decision to rebuild was made quickly. This summer I became curious how things looked like today, so I visited Dresden a second time.

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What may we forget, and how should we remember? Some of the temples and monuments that have been destroyed in the Middle East in the past few years were intended to last until the end of time by their creators. Arrogance, or trust in a protective higher power?

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We live in volatile times. A carelessly written email can haunt us for the rest of our live, while a mouse simple click can erase decades of work stored on a hard drive.

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If only we could attach an expiration date to everything we do, it would be easier to decide what to keep and what to let go.

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Golf Balls (Beach Balls Revisited)

A while ago, I showed how to visualize holomorphic self-maps of the sphere by drawing the pre-image of the standard polar coordinate system of the sphere (aka latitudes and longitudes). I mentioned that it should be possible to have these 3D printed, and here they are.

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They are printed with a gypsum printer, which is the only one have access to that can do color. That means that they are definitely neither suitable for golf or table tennis, nor for the bath tub. But I could use these for an exam in a Complex Analysis class. Each student gets one of these balls, and has to find out what rational function it represents. 


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The red lines (being the preimages of the longitudes) come together in the preimages of the two poles. Hence we can locate the zeroes and poles of the function. The only problem is that these pictures don’t distinguish between zero and infinity, nor tell they anything about scaling.

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They do tell about branching, i.e. the location of the zeroes of the derivative. For instance, in the blurry ball to left up above, we see a branched point where four of the red lines meet (instead of the expected 8 of the polar grid).


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Toxic City

There are (at least) two aspects of the DePauw Nature Park that I haven’t written about that make this place fascinating to me. One is the structure of the ground. 

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There is some weird flaky stuff that I haven’t seen elsewhere, but besides that, the ground is just more complex than what you typically would call Indiana Dirt

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I have waited to show this until now because, with early frost, everything gets even better. 

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The other aspect is the sound. In principle, this should be a quiet place (there rarely is anybody, at least not at my favorite hours). But there are birds, of course, and other noises, from factories and railroad tracks just not far enough away to be inaudible. Somebody should record this.

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Which brings me to another theme, that of ambiance in general. I have been listening to what is called ambient music for a while now, with increasing pleasure. Ambient music is not a well defined thing. It can just mean the incorporation of everyday sounds, or the questionable pleasure of background music. I like ambient music best when it distills everyday noise into something exceptional. Examples of that are Richard Skelton’s compositions (that are, in a good sense, very much down to earth), or, a recent discovery for me, Evan Caminiti’s recent music, including his new album Toxic City

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In photography (or even in art in general) there is the “classical” way to idealize the object — remove it from its context, isolate it, and even alienate it, in order to show a possibly artificially construed intrinsic beauty.

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Ambient art, in contrast, tries to show you how much there is without interference. We just have to look.

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That is a lie, of course. Whenever we show, we select. But selecting what we feel is worth seeing (or hearing) is very different from imposing a verdict on how things are on the viewer (or listener).

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When I was little, friends gave me as a birthday present a home made flip book that would show the deformation of the catenoid to the helicoid. That was a lot of work back then when you had to program all the 3D-graphics by hand, including hidden line algorithms. But I liked it to a have a physical object that would allow me to run my own little movie.

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Daumenkinos – Thumb Theaters, are they called in German. Today we see some snapshots from a high tec version of such a Daumenkino, attempting to get to the core of Boy’s surface (an immersion of the projective plane), about which I have written briefly before.

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The goal is to put a lid on a Möbius strip. The one we start with you will note is not just once twisted, but three times. I don’t know how essential this is to get an immersed projective plane at the end. I suppose it’s not, but makes things easier. Note that the strip has a single boundary curve, as expected.

The first two images show that Möbius strip, growing slowly. Below the first crucial step has happened: The growing strip has created a triple point, and intersection like that of three planes. But there still is only one boundary curve…

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We keep growing


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and growing:

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Another critical event: The boundary curve emerges completely into free air, i.e. doesn’t pierce through the surface anymore. Now it’s easy to close the lid:

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Mono Lake


Now that it is cold and gray outside, I like to travel a little back in time to pre-digital places. In 1993/94, while in California, I was at least three times at the iconic Mono Lake, that Mark Twain describes in his Roughing It as “one of the strangest freaks of Nature”.


Even though the water is extremely salty, some shrimp species seem to like it (Artemia monica, the Mono Lake Brine Shrimp…), and are in turn being liked by migratory birds.Berkeley588

I, in turn, found the Tufa rocks most fascinating. That they are visible and not underwater (where they originate) is a side effect of Los Angeles diverting water from the lake. 



That was still ongoing in 1993, but since 1994, after long legal battles, Mono Lake won and is now allowed to retain its water. So maybe the images here show Mono at a historic low.


In any case, this feels like home to me.


Why do we still teach geometry?  The constructions with ruler and compass were essential for the Egyptians and Greeks in order to accurately lay out large scale buildings with the only tool available (the rope). But today we have many other tools available, so there is no reason to confine ourselves to ruler and compass, unless we want to use them as a vehicle to teach the concept of proofs. That, however, is also in low demand, to the extent that graduating majors in mathematics are neither able to prove Pythagoras’ theorem nor to compute the distance of a point from a line.


I have been teaching a Geometry class twice now, and I am releasing the notes I wrote for the first part of the class into the wild. For this first part, I had set myself a few goals: I wanted to use only the most fundamental notions of geometry,  I wanted a plethora of interesting examples, and I wanted to be able to prove a substantial theorem. Finally, I needed to be able to give homework problems. 


The solution was to study incidence geometries, specializing pretty quickly to affine and projective spaces over arbitrary fields. So I did not develop axiomatic projective geometry, but rather taught the computational skills needed to quickly get to the geometry of conics in projective planes. This provided plenty of  exercises. Affine and projective transformations are intensely used in order to prove theorems or to simplify computations. The big theorem I prove at the end is Poncelet’s theorem.


The second half of the course? This deals with the two-dimensional geometries where we have circles: Möbius, Euclidean, spherical, hyperbolic. Emphasis is again on groups, and here in particular on reflection groups, proving Dyck’s theorem at the end. But I am not quite happy with the notes yet, so this part will have to wait.


So here is part I. Enjoy.

Notes on Geometry – Part I: Incidences






Quartet (DePauw Nature Park IV)

I like the days in late fall when Nature has gone to rest, but winter hasn’t arrived yet.

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We should do the same. Instead of denying the approaching darkness by putting up silly lights on dead trees, we should hesitate and contemplate the state of everything around us. 

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So I will conclude this year with posts and images that have more the character of still lives.


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It is time to pay tribute to what we will use for building: tree and stone.


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And to be thankful that time is still passing.

Six by Six

This puzzle consists of a 6×6 square board and 36 arrow cards with 6 arrows in 6 colors. The goal is to place all arrow cards on the board such that

  • the entire board is covered with cards;
  • each arrow points to exactly one arrow of the same color in the same row or column.


The example above shows incorrect placements of arrows: The yellow arrow cannot point to another arrow, the left blue arrow points to two different blue arrows, and it is not possible to add to the four purple arrows, while correctly placed, two more purple arrows without violating the rules.

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Indeed, if the six arrows of one color are correctly placed, the arrows can be connected to a closed circuit. The figure above shows two closed circuits of six arrow cards. Note that the circuit paths may well cross. The following puzzles have a few arrows already placed, and need to be completed. Below to the left is simple example that shows how to find the solution.

Ex 1

First consider the green arrows. The arrow at e4 points to the right and therefore we must have another green arrow at f4, either pointing up or down. As there is already a green arrow at f6, the new arrow at f4 must point up. This leaves us with two more green arrows to place. Because we already have three horizontal arrows, the remaining arrows must be both vertical, and point to existing horizontal arrows. The only possibility is to place the new green arrows at c6 and e2, as shown above to the right.

Now let’s consider the red arrows. The one at e3 points down, leaving no choice but to place a red arrow pointing left at e1. There are two more horizontal arrows to place, and the only possibilities are b5 and f3, see below to the left.

Ex 2

Turning to blue, there clearly needs to be a left pointing arrow at f1, and the two remaining vertical arrows need to go to a1 and b6.

For yellow we have only two arrows given, but there are not many free spots available. We first are forced to put a left pointing arrow at e6 and then a down pointing arrow at d6. The remaining horizontal arrows go to c5 and d4.

Ex 3

The purple arrow at d4 can only be reached by a right pointing arrow at a5, and the one at c3 only by an up pointing arrow at c1. Then the arrow at c1 requires a left pointing arrow at d1, and the final purple arrow goes to a3.

Filling the remaining spots with cyan arrows is now easy, giving the solution.

Below is a new puzzle to warm up:

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And here is a more difficult one. In both cases, there is only one solution.

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Everyday is silent and grey

Driving from Bloomington to Chicago takes four to five hours and is a pretty boring drive. This year’s highlight was a lone sign saying “Boycott Fox News”, a stark contrast to what we had last year at around this time.

There are many reasons to go to Chicago. This time I had the privilege to be needed as company, because my daughter wanted to see Morrissey.

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I am not very literate when it comes to popular music. The concerts I look back to with fondness had “stars” like Luigi Nono, Karlheinz Stockhausen, or Bhimsen Joshi present. No, I didn’t listen to The Smiths. But this is what our kids are for: To educate us.

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And Morrissey impresses. He says what he thinks not to be a clown, but to be himself, and he is still good at it. His band was dressed in T-shirts saying “This country makes me sick”. Morrissey was obviously very angry.

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The theater was sold out, with a very diverse crowd. A few rows away a group started smoking weed early on. Chicago. But they all had a good time.

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It was good to see this kind of solidarity across generations. It is the kind of angriness that brings people together.

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