Water Droplets

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Using a macro lens with 5-fold magnification is an odd experience. The usual, somewhat trivial “workflow” for taking a picture: Look-Frame-Capture doesn’t apply, because one doesn’t see what one might get until one is really close. 

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This time I was erring around in DePauw’s Nature Park’s quarry with its fascinating ground. How could I predict that the lump of greenness that has survived the recent cold spell here is up close a fully active miniature ecosystem, collecting and preserving water for nutrition and climatization?

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To see what you see here with the naked eye you’d need a magnifying glass. 

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The depth of field is of course abysmal, and I don’t usually have the patience to stack (at least) a dozen images.

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It’s good to know that there is a small world unfazed by the machinations of the big guys.

Ups and Downs

Lake Monroe is Bloomington’s water reservoir, and I imagine its waterfront properties are pricey. Having a large, perfectly situated peninsula becoming a Nature Preserve must have upset quite a few potential buyers.

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It took me a while to check it out. Traffic is low, mainly because parking is very limited. The only trail is straightforward, literally. It does a boring up and down along a modest ridge with an occasional view of the lake. If you go early, you will see herons and in years to come hopefully other critters who enjoy the lack of human presence. 

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So, while walking along, I was wondering what else one could do with a marvelous place like this. A single trail didn’t seem enough. Is this an opportunity lost? 

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At the end of the trail, one reaches a small beach where a few maple trees display their root system in a graceful dance. Changing water levels are hard at work to landscape.

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Let Nature do her work. I am happy to just watch.

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The Repository

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…

Fern Cliff Revisited (Ferns 6)

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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).

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This is maybe because it is not so easy to find, or that the humid Indiana climate recreates whatever grows quite rapidly.

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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.

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Some of the vertical walls are completely covered with soft mosses, liverwort, and other beautiful little things.

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Maybe we would be ecologically more reliable if our skin looked like that. Collecting water has become an art.

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Did I promise ferns? Here is an unusual one:

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Hidden Simplicity (Maybe-Ferns 5)

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.


Carpets (Foldables 4)

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!).MiuraTube

We begin with a corner type we call Double L

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Four copies of it (using reflecions) can be combined into a translationa fundamental piece like so:

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The tubes (of double length) emerge when we replicate this piece several times in both directions:

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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:MiuraWeave 4

So you can push this Miura Carpet to any of the four sides of a room.