War & War (Nordhouse Dunes I)

For many years we went camping to Nordhouse Dunes at Lake Michigan, and an episode of nostalgia made us revisit this place one last time before my daughter is off to college & life.

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In László Krasznahorkai deliberately cryptic book War & War, the hero György Korin is depersonalized: He just symbolizes a single function of our lives, namely delivering the past into the future, becoming the horizon between the below and the above.

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This happens concretely by carrying an old manuscript to New York, a place that the four inhabitants of that manuscript haven’t seen yet. These inhabitants are cryptic, too, bemoaning the loss of the noble, the great, and the transcendent, this causing also the loss of peace, so that the world now consist of only war & war. 

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Korin realizes however that delivering the manuscript is not enough, he feels the calling to complete it, to find an exit for its inhabitants. There are several attempts for this, one being by writing on water.

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This year, the dune grass that so gracefully used to create sand drawings, is now doing this in the water, thanks to water levels two feet above normal. The water itself leaves very temporary traces on the disappearing beach.

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Is it maybe the author’s dream that his protagonists keep writing the story?

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Noli turbare circulos meos! (Annuli VII)

In science, our goal should always be to present with clarity. Since the discovery of perspective drawings, a realistic representation of 3-dimensional objects has become almost mandatory. However, very often these objects have an appeal beyond their scientific truth which gets lost if its is shown in full clarity.



This blog has two series of posts titled “Spheres” and “Annuli” that both showcase images of simple 3-dimensional mathematical objects which deliberately forsake clarity in order to convey that other appeal. While accurate perspective renderings are used, the  perspective and textures are chosen as to emphasize the abstract aspect. 

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The example above shows a triply orthogonal system of surfaces. An easy way to create such a system is by taking a doubly orthogonal system of curves in the plane, revolve them about a common axis to obtain two families of surfaces of revolution that intersect orthogonally, and add all planes through the axis of revolution. For instance, we can choose two families of touching circles that pass through a common point, as above.


A single circle, rotated about the black axis, will revolve into a torus. To spice things up, let’s apply an inversion at a sphere centered at the intersection of the circles. This turns the tori into special cyclids like the one above, which all have the appearance of a plane with a handle. Using both a red and a green circle will invert-revolve in two such cyclids that intersect in a straight line and a circle:


These are still attempts of realistic drawings, but we already get the feeling that things aren’t completely evident anymore. For instance, the two cyclids above should be equals: but where did the corresponding red handle go?


Above is the same pair of objects from a different perspective. Now we can see the two handles and the intersection in a line, but where is the intersection circle? Also, where do we need to place the third surface family, which consists of inverted planes, i.e. spheres? The answer to that question is indicated below.


Other perspectives allow amusing variations:


For the top image, I have used several cyclids from each family, and several spheres, clipping them between two planes. To appreciate the image, all this knowledge might be irrelevant. To create it, it is essential.



We have learned simplicity,
we sing in the choir of cicadas


In 2004, the midwest of the USA became the region of the largest outbreak of biomass on the planet. Brood X emerged, the largest of the periodical cicadas, with a life cycle of 17 years.


They spend most of their life underneath as nymphs, going through several instars, and feeding from root juices of trees. Then, almost all on the same day, they emerge, and crawl up.


They molt into adulthood, which takes up to an hour. All this takes place above an abyss. If they fall, the soft wings will not unfold properly.


In the last stage, the wings unfold.


They hide and rest for a few days. This is supposedly the time they are most delicious.


Then, they tumble around, with their poor eyesight and clumsy bodies, causing harm to none, and begin their irresistible mating chant, droning sound patterns that move through the landscape like ambient music.

17 years underneath, for a few weeks of a single song. Who will question the meaning of life?

Time and Space

This has been an extraordinary summer. Weather wise, flooding rain falls were followed by torching heat, and now we are enjoying a dry summer weather that would be more typical for Northern Michigan. Time for a visit to the DePauw Nature Park, whose quarry enclosed space I would avoid on regular summer days. 

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It’s as green as it gets, promising a gorgeous fall coloring. Everything seems to take advantage of its given time and space.

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The abundance of vegetation creates patterns that are unusual for this place.

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More typical are the lonely little ones, like the young sycamore trying to make roots in the harsh ground,

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or the singular dandelion, gazing into our future.

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Imaginary Simplicity

Multiplication with the imaginary number √-1 accomplishes a counterclockwise 90° rotation, and that’s what today’s puzzle is based on. As casual readers will know, I enjoy simple things that rapidly get complicated. This puzzle is played on a rectangular checker board, with checker pieces in various colors placed on the squares. A move consists of rotating a checker about another checker by 90° either way. Below is the puzzle graph for a 3×3 board using two adjacent checkers, showing all possible positions and the possible moves indicated by edges that are colored by the checker you rotate about:

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It’s clear that in this case the two checkers have to stay together, and this poses no real challenge: Dealing with just two checkers is simple. Let’s add another one. Below you can see all possible six moves from the central position. You can either rotate about green or blue, but not (yet) about red.

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The puzzle graph in this case is connected and has diameter 7. You will notice that the checkers cannot change their parity. As simple count will tell you that there are therefore 80=5x4x4 possible positions or vertices. Below is a simple example how a puzzle could look like, with an optimal solution: Use as few moves as possible to get from the left position to the right one.


Naturally, things get trickier with more checkers and larger boards. Below is an optimal solution for a 4 checker problem that realizes the diameter 7 of the game graph (that has 240 vertices).


Finally, here is a problem on a 3×4 board with 4 checkers. The shortest solution takes 9 moves, and takes place on a game graph with 900 vertices. Having choices is hard, isn’t it?


There are many variations possible. One can, for instance, designate left and right handed checkers that can only rotate one way, making the puzzle graph directed. One can also turn this into a two person game by letting two players take turns. More about this at a later point.

Morning Song

Most of Indiana was either woodland or prairie, before the arrival of the white man.

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Imagine endless fields filled with tall grasses where you can get lost,

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where flowers spend all night to get ready

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for the morning, and where guest from the South are welcome.

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If you come early,

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when it is very quiet,

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you might even hear a voice from far away: Some mornings are better than others.

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Fold Me!

Last year, Jiangmei Wu and I worked on some infinite polyhedra that can be folded into two different planes. Today, you get the chance to make your own (finite version of it). This is a simple craft that, time and energy permitting, will be featured at a fundraiser for the WonderLab here in Bloomington. You will need 3 (7 for the large version) sheets of card stock, scissors, a ruler and craft knife for scoring, and plenty of tape. A cup of intellectually satisfying tea will help, as always. 

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Begin by downloading the template, print the first three pages onto card stock, and cut the shapes out as above.  Lightly score the shapes along the dashed and dot-dashed line, and valley and mountain fold along them.  Note that there are lines that switch between mountain and valley folds, but all folds are easy to do.

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The letters come into play next. Tape the edges with the same letters together. Begin with the smaller yellow shape, and complete the two halves of the larger blue shapes, but keep them separate for a moment, like so:

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Stick the yellow piece into one of the blue halves, this time matching the digits. Complete the generation 2 fractal by taping the second blue half to the yellow generation 1 fractal and the other blue half.

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This object can be squeezed together in two different planes. Ideal for people who can’t keep their hands to themselves. 

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The next 2 pages of the template repeat the first three without the markings, if you’d like to build a cleaner model. You then need two printouts of page 5. The last page allows you to add on and build the generation 3 fractal. You need 4 printouts. 

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Cut, score, and fold as shown above.

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Again, tape edges together as before. There are no letters here, but the pattern is the same as before. Finally, wiggle the generation 2 fractal into the new orange frame, as you did before with the yellow piece into the blue piece.

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Here is how they now grow in our backyard. If anybody is willing to make a  generation 4 or higher versions of this, please send images.

All these polyhedra have as boundary  just a simple closed curve. Topologists will enjoy figuring out the genus.