Mutual Attraction (Ohio IV)

The symbiosis of wood and stone is compelling, both in nature and in architecture.

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At first the timeless solidity of a rock appears to be no match for the organic softness of wood.

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But clearly the trees find stability, and hold on.

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Without the trees, this landscape would look barren, at best a symbol for an unachievable perpetuity, an abstraction, like a Japanese Zen garden.

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Sometimes, wood and stone seem to merge and become one, only different on the scale of time.

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Two Triangles (Solitaire I)

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This is the first in a potentially long series of solitaire games and puzzles that you can make yourself. Below you see two pairs  of quadrilaterals, each pair consisting of mirror symmetric copies. Print & cut them out, you will need  two sets to solve the puzzles. 

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These puzzle pieces consist of two right triangles that are glued together along their respective hypothenuse. I am using here triangles for which the short legs have lengths 1 and 8 for the first triangle and 4 and 7 for the other. Turning these triangles over gives the four possibilities above. 

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There are various ways these triangle fit together so that vertices meet vertices, and this creates many different ways to tile the plane.


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My choice of edge lengths is important, because 1²+8²=4²+7². This not only makes the hypothenuses of the two triangles equal, but also allows to fit the quadrilaterals together so that a vertex of one triangle meets another triangle along one of its edges, as the top figure shows.

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I don’t know whether there are such vertex-edge tilings of the entire plane, periodic or not. 

Below are three puzzles. Use two sets of the four quadrilaterals to tile the shapes below without overlaps or gaps. I hope they are not too easy.

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Gurgulio (Ohio III)

The word gorge is a gorgeous word, probably stemming from Latin gurgulio, which means throat. The Clifton Gorge in John Bryan State Park formed after dolomite deposited on shale, and the softer shale started to erode.

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Think about the armored skin of animals that protects softer layers and breaks with age or patience.

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The state park essentially surrounds this gorge, with long hiking trails following it on either side. Towards one end, the gorge narrows so that twigs from trees on opposite sides may touch.

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When the dissolution of the skin has progressed, what remains are large blocks of dolomite, that seem to be deposited magically into the landscape.

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They are like memories from a past that is otherwise forgotten, but still form the character of the landscape, without explanation.

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Time flows on.

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And erosion still continues, mercifully.

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Chance Visit (Ohio II)

Sometimes one needs a bit of luck and the right introduction to be admitted to a place.

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My second stop through Ohio was supposed to be John Bryan State Park, but my GPS told me at some point to leave the road. Obedient as I am I did, and ended up at a closed gate for the youth camp ground. When I started walking to explore where I actually was, I met Robert with his two dogs. We bumped elbows, and he explained to me the energy of the springs at this place, pointing vaguely in a direction where I should walk. 

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That turned out to be cross country for a few minutes, until I hit the well established trail system of Glen Hellen Nature Preserve, which I might have missed entirely had I found John Bryan State Park.

It’s indeed a curious place. Up above is a burial mound, and below the remnants of a bridge, maybe indicating that some chasms shouldn’t be crossed. The stubborn among us will keep trying.

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And it is indeed a place with many springs. The most iconic is Yellow Spring, whose orange-yellow color is due to large amounts of iron.

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The color persists for a while, causing ghostly reflections.

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Other springs feed small waterfalls. Four of them are named on the map, but there are more.

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The one below I discovered accidentally, by curiously stepping off the path. I was not the first.

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Erosion (Ohio I)

This blog post is about erosion as a design pattern, or about Terry Tempest Williams’s book of essays with the same name, or about the first stop at Caesar Creek State Park of my four day escape to Ohio, away from human interaction.

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We typically understand erosion as decay, as an increase in entropy, and we can observe it everywhere. This is one of the themes of Erosion, Williams’s very moving book. Erosion happens not only in geological matter, but everywhere: In the laws that should protect us, in our bodies, in our mental states.

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Resistance against this decay appears to be the essence of life. We lean against each other in support, until we break.

Caesar Creek State Park was a random pick for me on the way, and as such a disappointment. There is one long trail around the lake, which one cannot walk, because the bridge is under repair. What is a bridge that cannot be walked?

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But, as keenly observed in Erosion, there is another function of this decay: The creation of soil, of fertile ground for new growth.This becomes heartbreakingly intense in the chapter where Williams recounts the cremation of her brother after his suicide. This book is not an easy read, even for those of us who agree with Williams’s view of things.

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My return from the unwalkable bridge took me along the beach front of Caesar Creek Lake, which is not quite ready for building sand castles.

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But a closer look at the debris reveals that it is composed of older debris. Fossilesque, I would call that. Sometimes erosion takes a very long time.

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Read the book anyway. It will help you with your own, personal erosion. 

Corona Walk

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When you combine a pandemic with spring break with bad weather, you get this view from the top of the Atwater parking garage. Below is Lindley Hall. That the lights are on is unusual, but what isn’t?


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I continue my campus walk past the tiny Beck chapel to get to the IU cinema. 

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In front of it someone has adorned the pianist in a timely manner.

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The cinema itself is closed, like almost everything else.

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The way back takes me to Goodbody hall. Its terrace is eerily vacated.

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The only people I saw were construction workers and gardeners. Life has been reduced here to maintenance.

Pairs of Pants



One of the striking features of hyperbolic geometry is the possibility to construct shapes that are impossible in Euclidean geometry, and one of the most important such shapes is the right angled hyperbolic hexagon. Above is a tiling of the hyperbolic disk by copies of one of them.


They are useful to make pairs of pants by sewing two congruent such hexagons together to a single pair of pants, and then to continue sewing several such pants to eventually create coordinates of Teichmüller space. For that and other things, one needs to know that they actually exist, and the basic theorem here is that for any a,b,c>0 there is a unique right angled hyperbolic hexagon that has these numbers as the lengths of its odd edges.


Above is a short version of a proof of that theorem which I learned from Hermann Karcher. It employs parallel curves to geodesics in the hyperbolic plane. To see what they are, let’s switch to the upper half plane model. Geodesics (green) are vertical lines or circles orthogonal to the real axis (black).


A parallel curve (red) at distance d to such a geodesic is obtained by taking a point distance d away from it and applying hyperbolic translations along that geodesic to this point. These hyperbolic translations are simplest for the vertical geodesics, where they are just scalings with center at the end point of the geodesic. Therefore these orbits are also lines, or, in the case of circular geodesics, other circles ending at the same points as the geodesic (as follows using Möbius transformations)



Now let’s come back to the existence of right angled hexagons. Let’s start with a (blue) geodesic segment 𝛂 of length a. Two adjacent edges 𝛆 and 𝛇 begin at the end points of that segment and are orthogonal to it, but we don’t know yet how long they are. We construct parallel curves 𝛆’ and 𝛇’ to these geodesics, distance c and b away from them. Now take a geodesic 𝛅 that touches 𝛆’ and 𝛇’ in points B and C. The geodesic segment together with the perpendiculars from B to 𝛇 and C to 𝛆 completes the right angled hexagon with the prescribed edge lengths.

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To see the existence of the touching geodesic 𝛅, it is useful to switch again to the upper half plane, where we can assume that 𝛆 and 𝛆’ are straight lines. Then the existence of 𝛅 can be seen by looking at geodesic half circles 𝛅 near the end point and touching 𝛆’ and away from  𝛇’. Increasing their radius , monotonicity implies that there is a unique such half circle that also touches 𝛇’ (from the correct side). This then also establishes the uniqueness of the right angled hexagon, up to a hyperbolic isometry.