Berlin Alexanderplatz (10mm VI)

DSC 4478

In Alfred Döblin’s novel Berlin Alexanderplatz, the place of that name is being used to dramatically convey transformation: Franz Bieberkopf  is traumatized by the changes it has  undergone while he spent years in prison, and stands for the transformations he himself will undergo.

DSC 4458

Döblin’s novel takes place in the 1920s, and Berlin has undergo dramatic changes since. After the destructions of the Second World War and the division of the city, it was no longer the single city center. The architects of the Eastern part weren’t insensitive, they kept the space open and repurposable.

DSC 4473

Nearby churches were renovated and allowed other change to happen, later.

DSC 4570

After the fall of the Berlin Wall, many of the administrative buildings were taken down. The facade of the Palace of the Republic used to annoy the people of power with distorted reflections of the nearby cathedral. Not anymore.

DSC 4446

Radically modern buildings show that transformation is still possible. This leaves hope for Franz’s children.

DSC 4432

Hyperbolic Geometry

One of the most valuable human capabilities is doubt. Education seems to contradict this, early on we are encouraged to take certain things for granted. The trust in Euclid’s axioms for geometry was certainly universal and contributed to Immanuel Kant’s confidence that time and space are given to us as a priori.

Parallel 01

The ability to draw parallel lines, for instance, seems to be a given, and that this possibility is to a great deal responsible for being able to create realistic looking perspective drawings. What we can see with our own eyes must be true.


The discovery of hyperbolic geometry by Bolyai, Gauß, and Lobachevsky  is credited with overthrowing all this. If we are willing to accept that lines are not what they appear, but only have to obey all the other axioms of Euclid, then the parallel axiom does not need to hold. As mind boggling as this entire business is for the mathematician and philosopher, as irrelevant does it seem to be to the everyday person. After all, what we see is still true, isn’t it?


That there is a hard to explain esthetic appeal to that disk with its crazily symmetric patterns doesn’t quite justify the importance of hyperbolic geometry in contemporary mathematics either. Calculus at least is useful. That hyperbolic geometry makes its inevitable appearance whenever we study very simple things like the geometry of the circle or multiplication of 2×2 matrices, doesn’t really force us to talk about it, does it?


As flawed as our universal trust in the nature of space is our trust in the linearity of things. Double the income, double the happiness? Double the pain, halve the crime rate? It sounds too easy not to be true, and is often evidenced by the linearity of Euclidean geometry. Hyperbolic geometry is the simplest geometry where linearity fails and allows for dynamical systems with chaotic behavior. We have known this for over 100 years. We experience the effects on a daily basis, but prefer to ignore it.

Below are my class notes about hyperbolic geometry and incidence geometry, taught to undergraduates. Enjoy.

Notes on Hyperbolic Geometry (letter)

Notes on Hyperbolic Geometry (screen)

Notes on Incidence Geometry (letter)


Notes on Incidence Geometry (screen)

The Intimacy of Space (10 mm V)

Berlin and Bloomington have few things in common, besides their first letter B.

DSC 4416

Of more general interest is probably that both cities feature a building by Chinese architect I.M. Pei. I wrote about the Art Museum in Bloomington in an earlier post. Here you see the German Historical Museum in Berlin, or rather its extension.

DSC 4527

I would call this building an invitation to explore the esthetic possibilities of dysfunctional space. The helicoidal stairwell, it’s most prominent feature, connects only the second to the third floor and extends further without purpose to a non-existent fourth floor. It’s placed inconveniently at the (sharp) entrance corner of the building. Climbing these steps has as its main purpose to be climbing these steps. They are gorgeous.  

DSC 4521

The more functional connection between the ground floor and the first floor is a long ramp leading to the helix. Like everything else, it is pushed to the side, so that as much of the empty space of the building remains intact.

DSC 4540

What we see while walking this building are the structural elements that connect. Above is a view down into the basement level, reachable through the escalator or an angled stairwell (at the bottom).

DSC 4552

What I found striking and inexplicable is the harmony and balance between the playful round elements like the helix or the circular opening above, and the cornered, straight-edged, almost brutal structural components.

DSC 4554

It’s tempting to call these the male and female aspects of the building. No matter, it lives from the dialogue between the two. 


After the temperatures finally dropped to proper levels for January, it was time for another serendipity walk in the lightly frozen landscape.

DSC 4675

Usually I know when I have taken a decent photo. This time, I was not sure. When the warming sun came out, the reflections of the light and the doubly layered images of ice and ground beneath created unusual opportunities.

DSC 4684

Thawing is a violent process. This has never been made as visceral as in Andrei Tarkovsky’s film Solaris, in the scene where the visitor thaws.

DSC 4686

There, it’s the likeness of the alien that frightens. Here, the familiar shapes of leaves become alien when superposed with the fragile patterns of the ice that still covers them.

DSC 4689

There is a strange esthetic appeal in this violence, a desire to explode, and come to life.

Not yet. It’s January still.

DSC 4699

Marzahn mon amour

A neighbor and I exchanged books over the holidays (a forgotten art?). I gave her Christoph Ransmayr’s Arznei gegen die Sterblichkeit, and she returned the favor with Katja Oskamp’s Marzahn mon amour.

DSC 4348

Marzahn is a legendary suburb of Berlin I had never been to. The name triggered childhood memories of Frau Malzahn, the wonderful dragon in Ottfried Preußler’s even more wonderful Jim Knopf books.

DSC 4342

But this has nothing to do with Marzahn mon amour, nor do the pictures above, which show Alt-Marzahn, miraculously preserved among the Plattenbauten, the prefab buildings that provided a cheap solution to the growing housing problem of the former German Democratic Republic.

DSC 4385

Here is one of them, proudly announcing cosmetic studio at the entrance as if the entire building is that studio.

DSC 4365

And this is what the book is about: People living in these prefab houses, and being taken care of temporarily by the narrator, who works as a pedicurist in a cosmetic studio just like the one above (this one?).

DSC 4376

We learn to like them, the people and the buildings, maybe because they all have decided to cope with their large and small miseries by taking care of themselves, even if only symbolically.

DSC 4361

Most remarkable, however, is the insight of the narrator: That by stepping apparently down (in her case from struggling author to a pedicurist) one can in fact find happiness, and then by the way, write a charming little book. 


This morning I decided to replay the game Still Live that consists of walking around and taking pictures of things on the ground as they are. It is an exercise in acceptance.

DSC 2156

It was a crisp morning, nobody was out there that early on the first day of the new year.

DSC 2138

This time I made it even harder, by reducing everything to black and white, to dark and light.

DSC 2142

Maybe this modification of the concrete into something abstract is an escape to avoid comprehension.

DSC 2144

But by hiding the obvious, either the structural core becomes visible, or the underlying noise.

DSC 2145

Everything depends on what we want to see.