I started this little blog in late 2015 while being Director of Graduate studies in our department in order to keep sane. This proved (for me) to be very useful but a little irritating to the casual visitor who saw posts alternating between photography and a mild dose of mathematics.
This is about to change a little, because another project is taking my time. I call this the Minimal Surface Repository, a combination of blog and archive, that will vastly expand my minimal surface web pages. So my mathematical Monday posts here will become posts at the Repository, while the Friday posts will remain what this blog here was supposed to be: Thoughts about my inner state, accompanied by pictures, with an occasional game or puzzle thrown in.
So, please, head over, if you are interested in minimal surfaces and related topics. Otherwise, wait until Friday for more pictures from gloomy Indiana…
It is amusing to see that my own ways to look at things change faster than what I am looking at. 10 years ago, I took these pictures at the Fern Cliff Nature Preserve. Last week I decided to pay the place another visit, and fortunately, it hasn’t change much (in contrast to too many other things).
This is maybe because it is not so easy to find, or that the humid Indiana climate recreates whatever grows quite rapidly.
To the plants on the rock faces (bryophytes and ferns) these 10 years must seem like nothing. They have been around for millions of years.
So here we switch from wide angle to macro lens.
Some of the vertical walls are completely covered with soft mosses, liverwort, and other beautiful little things.
Maybe we would be ecologically more reliable if our skin looked like that. Collecting water has become an art.
Did I promise ferns? Here is an unusual one:
Mathematicians like to do things a little differently. An excellent example was the Mathematische Arbeitstagung, a yearly event held in Bonn, where the (mathematical) audience was asked to publicly suggest speakers.
Friedrich Hirzebruch would write the suggested names on the board (he sometimes misheard…), and then create a list of speakers on the fly. Sometimes they ended up with unexpected results. One year, Michael Barnsley was suggested, who had been working on a new fractal image compression method.
His talk was exciting for us graduate students, because we for once could understand something. The idea was to use special types of iterated function systems: Take a few linear maps that are all contractions, and use them to map a subset of the plane to the union of the images of that set under all the linear maps. This becomes a contraction of the space of closed subsets of the plane to itself with respect to the Hausdorff distance, and hence has a fixed point, which is again a subset of the plane.
It turns out that these subsets are highly complicated fractals, encoded just by a few numbers. For instance, all images on this page (except for the photo of Hirzebruch at the top) were made with just two linear maps, requiring 12 decimal numbers.
Barnsley claimed that he could reverse engineer this: Start with an image, and find a small collection of linear maps that would produce the given image very accurately. If true, this would revolutionize image compression. We went home and tried it out on our Atari ST computers and the likes. All we could produce were ferns, twigs, and leaves.
Paul Bourke has a nice web site where he explains how one can design some simple fractals, and has also some very impressive images of ferns using four and more linear maps. Below are the two simple maps used to create the polypodiopsida psychedelica above.
Our perception of reality is self-enforcing: We see what we are used to see. Artificially blurred, everything looks strange, ominous, threatening.
Still, we try to decipher an image and put it into the context of the familiar.
If this fails, we ignore it.
How much is out there that we ignore just because we never learned to see it?
The last (for now) example in this series of bifoldable designs is a woven carpet. Will create a doubly periodic polyhedron that consists of the Miura tubes below (which are almost 50 years old!).
We begin with a corner type we call Double L
Four copies of it (using reflecions) can be combined into a translationa fundamental piece like so:
The tubes (of double length) emerge when we replicate this piece several times in both directions:
Above is its most symmetric state. This carpet does not need to be rolled, it can be squeezed in both of its translational directions, as below:
So you can push this Miura Carpet to any of the four sides of a room.
What are blurred images good for?
Are they just there to cover up blemishes of reality or the lack of skills of the photographer?
Or is having more information always better? Shouldn’t at least something be in focus, always?
Or better, everything, with absolute clarity, so that nothing is hidden and no question remains?
Sometimes, I think, it is necessary to reconsider everything.
Whenever you show a mathematician two examples, s(he) wants to know them all. So, after the introductory examples of Butterfly and Fractal it’s time to make something more complicated. Jiangmei and I started by classifying all possible vertex types that can occur when you build polyhedra using only translations of four of the six types of faces of the rhombic dodecahedron (and make sure they attach to each other as they do it there). We found 14 different ones, and a particularly intriguing one is what we called the X:
The central vertex has valency 8, and we were wondering whether we could use it to build a triply periodic bifoldable polyhedron. It is easy to combine two such Xs to a Double X:
One can then put a second such Double X (with the order of the Xs switched) in front. Note that these are still polyhedra. Below are two deformation states of these quadruple Xs. We see that they are quite different.
So far, the construction can be periodically continued up/down and forward/backward. It is also possible to extend to the left/right, and there are in fact two such possibilities, allowing for infinite variations, because one has this choice for every left/right extension. They are indicated by the arrays below.
If you don’t have the time to build your own model, here again is a movie showing the unfolding/folding of a rotating Dos Equis.