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.

Pappus chain

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).

Pappus out

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:

Appolonius small

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.

0 1 6 + 0 1 2

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.