Past and Future

Once in a while it helps to go back in time a little. Indiana is a reasonable place for that, because during the Devonian period, some 390 Million years back, it was covered by a shallow see, a paradise for all kinds of critters small and big. They left us with plenty of fossils, and many of them are easy to find in stream beds.

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A famous place with a giant fossil bed is in the Falls of the Ohio State Park. The park itself is quite small and might come as a disappointment, as collecting fossils is obviously not allowed here. But one can take pictures.


This is somewhat serendipitous. I am not an expert, so I am completely clueless what the curious little sculptures on the rock bed are.

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Some might be rare, others just pieces of eroded trash. I don’t know.

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They are beautiful by themselves, and they set us into perspective: What fossils will we leave for casual visitors in 400 Million years? What will they think they see? Will there be a hint of civilization? What would we like them to see?

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Maybe the traces of a hand or a forgotten glove would be enough to tell: There was someone here who built.


Pappus Chains (Spheres VI)

Given two circles that touch at a point, fill the gap with a chain of touching circles. This is called a Pappus chain. In the image below, I show only two semicircles, and begin the Pappus chain with a circle touching the common diameter.

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Now take a circle with center at the point where the two given circles touch, and perpendicular to one of the circles of the Pappus chain we pick out. The inversion at this new circle takes the two given circles to two vertical lines, and the Pappus chain to a chain of circles between these two lines. The picked circle remains fixed. Below the selected circle from the Pappus chain there are precisely as many circles as to the right of the selected circle in the Pappus chain (four in the figure). Thus the height of the selected circle is determined by its diameter and its position in the Pappus chain. That, of course, will only excite the mathematician.

The same construction works in three dimensions. Take an arrangement of spheres between two vertical half planes, and invert them at a half sphere as shown.


The result is an arrangement of spheres between two hemispheres that touch at a point (where the spheres get really small).

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I thought this might be an interesting way to fill a dome. Standing in front of the entrance, with reflective spheres and reflective floor, might look like this:

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The Old People

About half way between the water fall and the White River, following the creek trail in McCormic Creek State Park,
there is a sharp bend in the creek, which makes the whole area a bit darker than everything else. In the middle of the creek one can spot a strange creature standing there and obviously waiting for us.

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At a closer distance, the creature reveals itself as the trunk of a dead tree, losing not much of its previous ominosity.


Its strong roots hold on to the icy water like the grip of a dead man’s hand.


The stump hints at the missing presence of a once magnificent tree. It is always what is not there that makes a place sacred.


This is a landscape that would best be illuminated by Paul Celan’s Fadensonnen. Elsewhere in the park, off the marked trails, a relative is still alive, barely, waiting as well.


The Little People


Spooky Yellowwood State Forest is home to the Bald Cypress, which produces roots that curiously protrude form the ground. These are called cypress knees, and it is rumored that they provide stability and oxygen in the swampy ground.


The truth is an entirely different story. When it gets dark and nobody watches, they begin to stretch and move.


Some stay be themselves, others meet in small groups.


They attempt to recapture familiar themes. Is this above the Holy Family? And that below Mary with Child?


Or do they just mock us? We will never know, as with brightening light, they return to their places and and pretend to be nothing but roots.

Frames and Borders

There are certain places I like to revisit from time to time like old friends whom I only meet once in a while.

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The interesting thing about this particular place is that it provides its own frame.

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In a photograph (like in any picture), the frame is the border between us and what we see.


Here, the frame consists of dead wood, horizontal and vertical, and allows the view into a changing and living nature before and behind the frame.


Taking such pictures is like an attempt to cross that border.


As in The Suspended Step of the Stork, the attempt fails, over and over again.


The Beauty of the Useless

Our perceived world is 3-dimensional, but even though most of us have a decently functioning stereoscopic vision, our ability to grasp the possibilities that space has to offer are quite limited. We rule space using box shaped blocks (houses). This is convenient, because it is simple and makes space accessible even computationally.

We surround ourselves with endless repetitions of familiar shapes, largely ignorant of the fact that there are many other simple ways to create and explore rather exotic shapes with an alien but compelling esthetics.

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The images from this page are all produced using quite simple formulas, using what are called harmonic functions.
They are related to minimal surfaces (soap films), but much more flexible.

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For the mathematician, the challenge is to find out how the algebraic properties of the formula are related to the geometric properties of the corresponding shape. This is largely done by experiment, to the surprise of many who don’t think Mathematics is an experimental science.

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At our fingertips we have infinite uncharted worlds to explore. We do not slaughter the natives, nor do we spend billions on super colliders or space probes.


Our discoveries are always fundamental, and useful only as a byproduct.


This sounds arrogant. In reality, it is just the belief that truly useful things have to be simple. This is our justification to explore simplicity for its own sake.




The fire pink is notoriously difficult to photograph. In the 3-dimensional wild nature, its five bright red petals catch the eye instantly and let us overlook annoying background or minor blemish.


Only after we have tamed its appearance on a 2-dimensional photograph, the defects become immediately apparent. The uniform red shows the tiniest specks of dirt, and little tears in the petals that went unnoticed in nature become major issues. Even its own pollen becomes a nuisance in the photo.


Of course the right choice of light, depth of field, and post processing help. But I am still waiting for the perfect specimen for the perfect shot.


Wood and Stone

Even more than the near Shades State Park, Turkey Run State Park offers a maze of narrow canyons filled with remnants from the retreating glaciers of some 20,000 years ago.

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A common theme is the presence of wood and stone. Most of us are surrounded by their shaped presence more or less permanently, but here we can watch them grow and decay in their raw and untamed state.

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This place has something special at any season. In early spring, the abundant vegetation is still dormant, and the damage done by the melting ice and snow has not been cleaned up yet.

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This will just look like devastation to most, reminding us that building with wood or rock is, in the long run, nothing but building on sand.

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Occasionally, there is a view that seems to contradict the chaos. While such views are nothing but rare byproducts of the greater erosive randomness, they still remind us that there is purpose, as long as we pursue it.

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Simple Beginnings (Spheres V)

The simplest way to arrange spheres in space is to use the cubical lattice. This is the obvious generalization of the checkerboard, and it lends itself naturally to a coloring with two colors such that neighboring spheres are differently colored. While this is not the densest sphere packing, it will be pretty dark inside.


Leaving out the spheres of one color, painting the rest with most of RGB color space creates the following arrangement of spheres, and makes enough room for light to get through.


Now imagine yourself inside of it, and all spheres being reflective in addition to being colored. The formerly simplistic object becomes a dazzling fractal-like maze.

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The original bicolored sphere packing is related to a packing of space by octahedra (one for each orange sphere).
Two octahedra share then at most an edge, and the gaps can be filled with regular tetrahedra of the same edge length.


Minkowski discovered that octahedra can be packed much more densely. The gaps can still be filled with regular tetrahedra, but their edge length is only one third of the edge length of the octahedra.


First Flush

Today, I took possession of a medium sized box from India.

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It contained lots of little nicely labeled bags full with tea leaves.

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One of my yearly delights that few understand is the arrival of the new harvest Darjeeling, the main ones being the First Flush (March) and the Second Flush (June). Here we are looking at teas from a dozen or so tea gardens. This is to the tea drinker what vineyards are to the wine drinker. And similar to wine, teas vary in quality and quantity from year to year.

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Even though the British are responsible for stealing seeds of tea plants from China and planting them in India, most of my friends from England have a hard time recognizing the dried leaves as tea. Most of them are used to the dust that is now cheaply produced in Africa. Already smelling the dry leaves is wonderful.

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Then, of course, there is the science of steeping tea properly. Temperature is important; for good Darjeeling the water should be boiling or near boiling. More difficult is the choice of the right water. The tap water in most places I have lived completely ruins the tea. Filtering often helps somewhat, but bottled water is better, and finding the right one is not easy. Luckily, the local tap water in Bloomington is excellent for tea (except when a summer draught causes it to taste muddy). For reasons I haven’t been able to find out, tea doesn’t get bitter here.
This is limestone country, and the tap water is surface water from artificial reservoirs.

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Finally, it is essential to give the leaves some space to expand. After all these preparations, I am rewarded with several delightful cups. Strange that most people are clueless about all this.

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