That my father taught music in high school influenced the development of my musical taste to a large degree. My initial resistance to classical music resulted in the common appreciation of popular music, of which I still approve groups like The Cure and Alan Parson’s Project but disapprove others which I (in shame) will not mention.

This attitude shifted after an organ recital that featured a piece by Olivier Messiaen.



I secretly searched my father’s vast vinyl record collection, and was delighted to find only one record, namely the Visions de l’Amen, played wonderfully by Yvonne Loriod and Messiaen himself.

This was the beginning of an attraction to contemporary music. I grew up listening innumerable times to an hour long piece called Computerimprovisationen auf dem Hybrid 5 that I had recorded from public radio using my father’s reel to reel tape recorder. I would love to get hold of a recording of that piece, but alas the internet is not almighty, and I have not been able to track it down.

After I moved to Bonn I kept exploring. I still think the main entrance to contemporary music are Bartok’s six string quartets: If you are able to thoroughly enjoy these, you are ready for everything else.

The Bonn-Cologne area was well suited for developing taste. The Cologne Philharmonie, which is underneath the fountain in the picture below, offered

Cologne 1

not only the standard classical fare, but also frequent concerts by composers that were not only alive but in fact present. This gave authenticity and a sense of being in presence of the creative act, which I find stimulating.

What makes me post this? A news report, that in that same place, in March 2016, a cembalo recital by Mahan Esfahani that featured besides the Bach family also Steve Reich, was cut short by an unruly audience that apparently could neither deal with Reich’s 50 year old Piano Phase nor with the ethnicity of the performer.

Cologne 2

Cologne has something to offer for every taste. Why offend others?

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.

Quarters 01

To make things visible, we have to remove some of them, and a natural choice is to remove every other arc, like so:

Quarters wiggle 01

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.