## Bianchi and Hopf (Annuli V)

When you take a torus in Euclidean space, it will always have points of positive curvature and points of negative curvature, but the average of the curvature will be 0.

This limitation of Euclidean space disappears in the 3-sphere. Luigi Bianchi more or less completely classified all flat tori in the 3-sphere in the late 19th century, paving the way to global differential geometry.

Simple examples can be constructed using the Hopf fibration: If you take the preimage of a simple closed curve in the 2-sphere under the Hopf fibration, you obtain a torus in the 3-sphere that is intrinsically flat.

When you start with a circle in the 2-sphere, the result will be a torus of revolution (or a Dupin cyclide), which corresponds to a rectangular torus. If you start with more general curves like the spherical cycloids that I have used here, you will get tori that appear twisted.
This is because these tori will correspond to non-rectangular tori, as can be verified using an elegant formula due to Ulrich Pinkall. The side view below gives access to the warped core of these tori.

The image below shows the view from somebody standing inside such a Hopf torus, with an almost perfect mirror as a surface, and four differently colored light sources. This comes pretty close to my ideal of abstract 3-dimensional art. If you know what you are looking at, you can discern the same warped core as up above, and its reflections on the warped outer parts of the torus.

## Wesseling at Night

Wesseling is a scenic industrial area about half way between Bonn and Cologne.

The mostly functional architecture and perpetual construction is rarely as amusing as in the picture above. The time to be there is at night, when the architecture of metal and concrete is replaced by a much more fundamental architecture of light and shadow.

In the nearby harbor, large cranes appear to be asleep. Are they dreaming of electric sheep, too?

And then there are the relics from the past, like this barely recognizable wind mill.

With increasing darkness, the film grain takes over. Is this how Georges Seurat would have painted this? I wish.

## Spherical Cycloids

The cycloids generalize nicely to curves on the sphere. They can be physically generated by letting one movable cone roll on a fixed cone, keeping their tips together, and tracing the motion of a point in the plane of the base of the rolling cone. Like so:

Varying the shapes of the cones will gives differently shaped cycloids, most of which will not close. When they do, they have a tremendously appeal (to me) as 3-dimensional designs, like this tent frame

or this candle holder:

In a future post I will use the following curves for another construction. Each is without self-intersection

and together they form a nice cage that from the side has an organic appearance.

I’d be interested to learn how such objects could be manufactured, say as pieces of jewelry. How does one bend metal tubes accurately?

The images are created using explicit formulas for the cycloids, but rendering approximations of spherical sweeps about cubic splines in PoVRay.

## The Woman in the Dunes

While living in Bonn, I often went to the local art house cinema, the Brotfabrik (bread factory), not knowing what to expect. I was often rewarded with surprises, but few were as impressionable as watching The Woman in the Dunes by Hiroshi Teshigahara.

The next morning I went to a bookstore (yes, it’s that long ago) and bought Kobo Abe’s book with the same title. The book, while still worth reading, pales compared to the film, which is still sticking with me, in particular when I visit dunes

The pictures here are from Oregon in 1994, when I visited Christine and Tom.

I paid them back their hospitality by taking these pictures.

And no, we did not try to reenact the film. But the intensity of the landscape almost too easily distills the personality of the visitor.

If I ever feel like emotional cleansing, I will walk the Oregon Dune Trail.

## What to Keep

I have often been trying to capture personal time in this blog through old and new photographs. What you see above, is a recent (like 20 minutes ago) photo of an old toy of mine. I received this early edition of Spirograph when I was maybe 9 years old. You can now purchase a 50th anniversary edition (without the tasty pins …).

This is something that I (and my daughter) have used intermittently over all these years, and it has acquired a meaning for me way beyond its mere presence. Already back then I cherished it so much that I kept the products in the box. So this is, well, an ancient artifact:

The lavishly illustrated instruction manual promised perfection that I never achieved. Too often one of the wheels started sliding instead of just rolling, or the pins didn’t quite hold. What counts, however, is the process. We are, truly, not interested in the ideal, the mathematical perfect curve, but in the process of getting there.

The curves that one can make with Spirograph are called Cycloids. You can get them abstractly by tracing a point on a wheel that is rolling along a curve. In its simplest form, you roll a circle along a line,

and you learn that these curves can be found on the icy Saturn moon Europa, or as geodesics in the upper half plane when using the Riemannian metric 1/y ds (which is not quite the hyperbolic metric, of course). The ancient ones used them to model planetary orbits when popular belief pinned man into the center of the universe.

As my early Spirograph experiments show, the results make nice designs. Using contemporary software like Mathematica allows you to create these to perfection, you think? Unfortunately, plotting the true cycloids will result in images that are either inaccurate (not enough anchor points) or difficult to manipulate in Adobe Illustrator (too many anchor points). So, to make this:-:,

I replaced the cycloidal arcs between intersections by cubic Bezier splines that have the same curvature as the cycloids at their end points. Again, this was just to find satisfaction in the process to approximate the ideal.

## Double Exposures

The Sieg is a tributary of the Rhine northeast of Bonn. The word Sieg means victory in German, but (wikipedia tells me) the name of the stream derives from the celtic word sikkere (fast stream), as does the name of the French Seine via the related Sequana. This must be flattering for the Sieg.

A slightly elevated dam next to it gives the opportunity to extended bike rides. I have written before about the area here, and I am revisiting the place now, as I revisited it often in the 1990s.

The dam also provides an excellent perspective on the trees

or the power line masts.

The picture above was made using a now obsolete technique, the double exposure. I used to experiment with it quite a bit,
but gave up on it when doing this became more or less trivial in Photoshop. It is disappointing to see how the creative possibilities of multiple exposures have become reduced to automatized photo stacking with the goal to increase the dynamic range or depth of field.

## Choice and Fate

Jigsaw puzzles are terrible. They tap into our subconscious desire to complete tasks even if they are pointless, the reusability is minimal, and they offer next to nothing for the creative or just inquisitive mind.

When in one of my former lives I needed a creative activity for young children that would encourage them to view themselves as part of a group, I designed the anti-puzzle above. It consists of only one piece, which is blank. Each child would receive a large printout of the tile, cut it out, decorate it with something personal, and put it on the wall.

There is no difficulty putting the pieces together. Everything fits, and you cannot make mistakes. The only choices that are left are the designs of the individual pieces, and the place on the wall. I can imagine grownups could use these for brainstorming, post-it style.

In a later part of that life, I recycled the idea for older children. Here, there are three different snowflake shaped jigsaw pieces (which I had cut out in large numbers and many colors using a die cutter). This turned out to be surprisingly difficult, because the pieces appear to allow you some choices. However, if you want to fill larger regions without gaps, it will always look like this:

The hexagonal lattice is something our squared brains have a hard time to adjust to, apparently. Still, choices can be made by selecting the colors and shape of the design.