Coal and Gold (Columbia Mine Preserve III)

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After two slightly misleading posts about a facility (inside and outside) near the Columbia Mine Preserve, the time has come to visit the actual place.

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Large open spaces (either prairies or lakes) allow to look far into the distance in all directions.

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It conveys an almost paradoxical state of mind: Being small and unimportant in this vast landscape, but also (subjectively) being at the center of it.

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Being there in the late afternoon gives the opportunity to experience another contrast. Just before the wintry gray turns into the black of the night, the sun makes one last effort, just before it hits the horizon.

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Then, for a short moment, the black and the gold coexist, and the limiting horizons become an illusion.

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Mirrors and Diamants

In the 1970s, the German fountain pen manufacturer Pelikan ventured into the budding market of authored board games with a series of games in a well designed format. The market wasn’t ready, and some of the games had serious issues with their rules. I bought three of them, and only one was a keeper: Diamant, by Andrea Steyn.

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You got a game board (advertised as 40mm thick…) to hold 12 silvery plastic mirrors and 16 small diamonds in 4 colors, as well as a set of goal cards. Players would take turns placing mirrors on the board, or later moving them. Then they could send (virtual) light beams from their side of the board, having them reflect on the mirrors to eventually hit one of the diamonds, which the player could then take in order to work towards completion of their personal goal card. 

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Above is a top view of the board, showing mirrors and color coded light beams. The numbers at the border indicate how many mirrors are hit by the light beam, starting/ending at that number.

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By discarding the colors and the mirrors, you can turn any mirror configuration into a puzzle, like the one above. A good strategy is to place mirrors that give the correct small numbers. While not forced, a first step towards a solution might look like this, dealing with the 2s:Ex4 4a

Now you can see which of the 3s have to connect, and from there the solution is easy to find. Below is a much harder puzzle on a 6×6 board. My solution uses 15 mirrors of each kind.


I don’t know how (computationally) hard these problems are, but they are obviously NP (non-deterministically polynomial): It is trivial to check whether a solution is correct. There are many variations of this simple puzzle (building in refraction, for instance…), and cute little math problems one can ask about the numbers that can show up at the border. Maybe I’ll come back to this in the future.


The Dark Tower (Columbia Mine Preserve II)

After looking at the mining facility near the Columbia Mine Preserve from the outside last week, now it’s time to step inside.

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This is already the second floor, from a total of six. Thanks to the broken windows, the wind has done a decent job cleaning the place.

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Moving up. This feels like one of these dungeon computer games where you have to deal with cute monsters on the way up (or down). I am pretty sure I know where the undead from the three (!) cemeteries I passed on the way spend their free nights.

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Further up. It also reminds me of Snakes and Ladders. One misstep, and you have to start climbing all over again, if you can.

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The eeriest part of the place is the sound. Birds have conquered it, and the sounds they make are surprisingly close to human chatter. Maybe this place is some sort of temple for them.

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It also feels like I am an exploring some alien space ship. I have absolutely no clue what these enormous machines were used for. 

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Not only birds have left their stains. Monsters, undead, animals, aliens — what do we fear most?

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Down again, unharmed. Two decades ago, this place was busy with people who worked there. Where are they now, what are their stories?

Hyperbolic Geodesics (Geometry and Numbers II)

One of Euclid’s axioms states that lines can be extended indefinitely. If we have to be content with a finite canvas, like the rectangle below, we have to resort to a trick to keep lines going when they hit the boundary of the canvas. One such trick is to allow the line to re-enter the canvas at the corresponding position on the opposite side of the canvas. Those of us who have been exposed to asteroids in their dark past are familiar with this. Lines with rational slope will thus look like this:

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If the slope is irrational, they will become dense, i.e. come arbitrarily close to any given point on the canvas. In either case, all segments of a given line are parallel. This changes when we switch to hyperbolic geometry, represented by the upper half plane, where lines (now called geodesics) are half circles perpendicular to the boundary, or vertical lines.

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Above you see three such geodesics, forming the boundary of the shaded region, which will be our canvas. This canvas is the classical fundamental domain of the modular group. If we want to follow hyperbolic geodesics in this canvas, we have to explain what to do when a geodesics hits one of the sides of this infinite triangle. This is easy for the left and right vertical edges: If the green geodesics exits at the right hand side and becomes purple (due to lack of oxygen), it is translated back to the left. This translation by -1 is in fact not only a Euclidean congruence, but also a hyperbolic one.Artintrans2 01

When a geodesic tries to exit at the circular bottom, it is rotated back, using a hyperbolic rotation by 180º about the point √-1, which is in complex numbers given by z↦-1/z. That’s a bit harder to visualize for our Euclidean eyes. One way to think about it is to find the point where the green geodesic becomes purple, take the corresponding point on the other side of the black dot on the bottom red half circle, and continue with a purple geodesic up into the shaded region by making the same angle as the one you made when exiting.

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This allows to draw hyperbolic geodesics just like we did in the rectangle with Euclidean lines. Below is a more complete picture that shows how much more complicated or chaotic this is becoming when we keep extending the geodesic. The numbers help to identify end points of segments. How complicated can this get?


The two operations that allow us to continue a geodesic, namely z ↦z±1 and z↦-1/z are exactly the operations that we used last week to find the continued fraction expansion of a real number. This seemingly far fetched connection points to a deep link between number theory and geometry: Take a geodesic in the upper half plane, and look at its left and right end points a<b on the real axis. We will limit our attention to the case that -1<a<0 and 1<b. Write both -a and b as continued fractions, this gives two sequences of positive integers  a₁, a₂,… and b₀, b₁, b₂. We combine these into a single bi-infinite sequence …b₋₂,b₋₁,b₀,a₁, a₂,… which we denote by cᵢ.

Now it turns out that continuing the geodesic across the edges changes the sequence cᵢ either into c₋ᵢ₋₁ or into cᵢ₊₂. Either operation represents a mere shift or flip of this sequence. This leads to a remarkable theorem of Emil Artin from 1929: There is a hyperbolic geodesic on this canvas that comes any given hyperbolic segment on the canvas arbitrarily close. So this geodesic is not only dense, but dense in all directions simultaneously.

To see this (a little bit), it suffices to find a suitable bi-infinite sequence cᵢ of integers encoding that geodesic. Take as cᵢ a sequence that contains every finite positive sequence of positive integers as a subsequence. Then, for a given geodesic segment, find its bi-infinite sequence, and truncate it at both ends to get a finite sequence. By truncating further out, we obtain better and ebtter approximations of the given geodesic. As every finite sequence of positive integers is contained in our sequence  cᵢ, we will (by continuation) eventually find a segment of the encoded geodesic that is as close to the given segment as we desire.

This theorem of Artin is at the beginning of the study of both geometrical dynamical systems (the geodesic flow) and symbolic ones that are related to number theory and exhibit a chaotic behavior that is not apparent in Euclidean geometry.



Come in Without Knocking (Columbia Mine Preserve I)

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Spring last year, on my way back from New Harmony, I made a small detour to the Columbia Mine Preserve. The Vigo Coal Company mined the area in the 1990, then filled the holes, and let it sit. The Sycamore Land Trust acquired the area, turned it into a nature preserve, which is now part of Patoka River National Wildlife Refuge.

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Last year the early warm weather didn’t encourage any good pictures, so I decided to return a bit earlier, to catch the gloomy Indiana winter. When I entered Patoka River National Wildlife Refuge into my GPS, it took me to a dead end just outside the refuge, but I passed this wonderful relic on the way.

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About six floors tall, this structure was apparently used to do something to the coal before it was used to enrich our atmosphere with carbon dioxide.

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I am also clueless about the purpose of this truck, and why it looks so unhappy.

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This time, the door was missing, so again I couldn’t resist the temptation. There was quite a bit to explore inside, so I leave this as a teaser for next week:

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What is a Number? (Geometry and Numbers I)

The moment when humans made the abstraction from a set of objects to its cardinality and thus discovered counting is lost in history. There are other moments of similar impact that we know more about. Today, we are so familiar with numbers that we often forget that they are used to measure quantities and even ignore units, confusing distances and durations.


For the early pre-Aristotle Greek mathematicians, lengths were not numbers at all. Numbers occurred as proportions, as ratios of lengths (or durations). One segment could be say twice as long as another segment. More generally, the Greeks called two lengths commensurable if their ratio is, in modern terms, a rational number. They would detect this by fitting one number, like 30, as often as possible (once) into another number (43), take the remainder (13), fit that in the second number (30) as often as possible (twice), take the remainder (4) etc. etc. What emerges is the continued fraction above, terminating eventually, because the denominators get smaller and smaller.


If the lengths are not commensurable, the continued fraction becomes infinite, like the one above. For the Greeks, this expression was essentially an algorithm. An infinite fraction is a mind boggling thing. How does one even compute it?


Geometrically, this can infinite continued fraction arises by comparing the side length 𝝋 of a regular pentagon to a segment of length 1 of its diagonals. Simple similarity of triangles tells us that 𝝋=1+1/𝝋. Rewriting this once leads to 𝝋=1+1/(1+1/𝝋), and if we keep going a little while longer, we arrive at the infinite continued fraction above. This reproduces how the Greeks proved that the Golden ratio 𝝋 is irrational (if it was rational, the continued fraction would be finite).


Similarly, the above dissection of a unit square into a rectangle shows that (√2-1)(√2+1)=1. This is arithmetically easy, but the concept of a root of a number didn’t make sense to the Greeks in the early days. This equation is the essential ingredient to prove the continued fraction expansion of √2 (and thus its irrationality).


Of course, the standard proof by contradiction that is taught today (and which probably goes back to Aristotle) makes the cumbersome process of finding continued fractions cumbersome. We will see next week that they still serve higher purposes.