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.

Exposed (10 years ago)

The word Steingrund appears in the title of a post that recollects a visit to Desolation Wilderness 25 years ago. 

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Reminiscing today about a visit to Turkey Run State Park 10 years ago let’s me use another word from the same poem.

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The word exposure includes visibility, fragility and presence, and the ominously dark landscape doesn’t seem to convey this, until you notice the cracks, traces of violence that happened here many thousand years ago, unmeasurable for us.

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Patient streams have smoothed the rock and created paths that can be walked best upstream, against time.

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Trees hold on to the rocks with roots like fingers for decades, while unknown plants seem to be ready to flee any minute.

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Napoleon’s Theorem

Coming fall semester  I’ll be teaching our geometry undergraduate course, and to prepare myself a little, I am going to try some of the ideas I’d like to explain. One emphasis is on simplicity. If we just concern ourselves with points in the (say Euclidean) plane, a first fundamental question is whether they can be in an interesting position, and if so, how we can tell?

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There is (almost) nothing to say here for one or two points, but for three points it gets interesting. Three points can be collinear, a condition from projective geometry, about which I know what to say. Another natural choice is to place the three points at the vertices of an equilateral triangle

That is already a fairly advanced concept. How to you explain what an equilateral triangle is to your friend’s children in elementary school? As a full orbit of a cyclic group of order 3? That is, in fact, the simplest definition I know, avoiding conceptually much more difficult measurements of lengths or angles. It also has the advantage that it generalizes to other geometries.

How do we tell? An elegant answer in the spirit of the orbit definition can be given be denoting the vertices A, B, C as complex numbers. They lie at the vertices of an equilateral triangle if and only if A+𝛇B+𝛇²C=0, where 𝛇 denotes a third root of unity: 𝛇 = (-1+i√3)/2. This is easy to see, because it holds for your favorite equilateral triangle, and all equilateral triangles are similar, i.e. differ by a complex linear transformation which changes the left hand side of the test equation by a nonzero factor. I know I will have to say a little more, but not much.

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What is this good for? One amusing application is Napoleon’s Theorem:

Let ABC be any triangle. Construct equilateral Napoleon triangles CBA’, ACB’ and BAC’ outside of ABC. Then the centroids A”, B”, C” of the Napoleon triangles form another equilateral triangle.

The proof is very simple now: We just express the centroid A’’ of CBA’ as (C+B+A’)/3, and likewise the other two centroids as B’’=(A+C+B’)/3 and C’’=(B+A+C’)/3, and apply the test.

Could there possibly a better proof? Yes and no. Here is one that is more a revelation than a proof.



The first key observation is that there is a tiling of the plane, using the given triangle and the equilateral Napoleon triangles, as shown above. This tiling becomes periodic with a hexagonal period lattice.


This allows to overlay the Napoleon tiling with a tiling by equilateral triangles that have their vertices at the centroids of the Napoleon triangles. 

The reader will find it amusing to fit this all together. How general is all this? For instance, can we do this in a plane over a field without a third root of unity?

It is not clear whether Napoleon has actually proven this theorem. He was interested in Mathematics, and discussed science with Lagrange and Laplace. The first known written appearance is an article by W.
Rutherford in 1825, which doesn’t mention Napoleon.



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A few weeks ago, my daughter brought home several of these. A waterplant. Neither she nor I know what it’s called. She says she downloaded it. Language. Reality. 18 years ago I expected she’d been driven me mad with new fashionable forms of body modifications. Our children are there to surprise us. 

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Let’s have a closer look. The ant is there to eat to show us the scale. This is the stuff above water. Leaves.

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And here are the roots. We first kept the plants outside in the shade in tap water, which they didn’t like. Now they are in the sun in tubs full with rain water, which they seem to love. The roots have grown immensely, making it stay.

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They consist of long, pale strands with many smaller, almost translucent tendrils branching out. I had never looked at a clean, delicate root system like this before.

Do mermaids have hair like this?

New Harmony State Park (New Harmony IV)

We return for a last time this year to New Harmony. This time we visit the nearby state park, my original motivation to travel to the southwest corner of Indiana.

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The park is rough and unkempt, without must-see spots. Instead, you get largely untouched woodland where you have to find the subtle beauty yourself.

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May Apples play with blurred highlights as if they have been waiting for me, and somebody has left a message in the dried river bed. Unlikely, but we can dream.

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On this Friday morning a good month ago, I am the only visitor. Below once again the Wabash river, shortly before he finds oblivion in the Ohio.

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The landscape doesn’t quite feel like an Indiana landscape anymore. This is already the South. The wonderfully braided bark of the Pecan tree cannot be seen much further north.

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There is somebody else, after all. Patient Cassiopeia, waiting for the years to pass.

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The Final Chapter

I like taking pictures of people, but rarely post them. Today will I make an exception, because Jessica and Matthew are celebrating their 10th wedding anniversary, of which I had the honor and pleasure to take the pictures. Their first decade…

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Taking portraits is difficult. Heinrich von Kleist finishes his essay about the marionette theater with the dialogue:

“So we have to eat from the tree of knowledge a second time in order to achieve the state of innocence again?” – “Yes, indeed, and this will be the final chapter of the history of the world.”

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People are nor landscapes. You can treat them like that when they are unobserved to snitch the occasional good photo. Most people freeze when they become aware that somebody is taking a picture. 

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Matthew’s parents are a rare exception, the personified confidence in themselves and the world.

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Then, of course, people also don’t hold still, as do most landscapes. So one has to be vigilant and be quick.

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Happy Anniversary!

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