Arrivals and Departures

This is an unusual post, marking arrivals and departures.

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Even worse, the sea creatures on display appear to have nothing to do with that theme. Let me explain. One of the arrivals is that of my daughter arriving at the critical age of 18, and one of the departures is hers to college in California. This provides a first link: The pictures are from the Monterey Aquarium, which we visited last year.

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When I see these astonishing creatures, I am inevitably reminded of Denis Villeneuve’s film Arrival, a rare example of an adaptation that works independently and as well in its own way as the source, here Ted Chiang’s The Story of Your Life. The departure I will associate with this is that of the composer of the wondrous film score, Jóhann Jóhannsson, who left us last year, too early.

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Arrival and departure sound like beginning and end, joy and sadness. This is treacherous, because each departure is a departure to a new arrival elsewhere. Arrival and departure are like a single contraction of one of these jellyfish. What you perceive depends of where you are: inside or outside.

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More important than arrival and departure are the stories that are framed in between, the mysterious creatures that propel our lives forward or bring it to a halt.

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I am looking forward to hear more.

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The Fence Had a Hole (New Harmony I)

The soul pattern of a bridge is a straightforward one, we use it to cross from one state into another. I have mentioned related patterns before, that of multiple crossings and that of the arch. Today we talk about the pattern of the closed bridge.

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This is the Harmony Way Bridge over the Wabash River in the town New Harmony (about which we’ll learn more next time). The bridge opened in 1930 and was used as a toll bridge, and was designated as structurally deficient, and has been closed since 2012.

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It has been clearly adorned with warnings, as you can see. But somebody cut a hole into the fence, and those who know me can guess what happened.

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Which brings us to the point of this pattern: A closed bridge can be used for crossing, but there is a price to pay. You don’t cross such a bridge casually, you hesitate.

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Angelopoulos’ mindshattering film The Suspended Step of the Stork distills this moment of hesitation. What happens in us when we consider to leave, to cross over? 

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Standing there, looking back, and looking forward can last an eternity. Don’t do this often.

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Daumenkino

When I was little, friends gave me as a birthday present a home made flip book that would show the deformation of the catenoid to the helicoid. That was a lot of work back then when you had to program all the 3D-graphics by hand, including hidden line algorithms. But I liked it to a have a physical object that would allow me to run my own little movie.

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Daumenkinos – Thumb Theaters, are they called in German. Today we see some snapshots from a high tec version of such a Daumenkino, attempting to get to the core of Boy’s surface (an immersion of the projective plane), about which I have written briefly before.

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The goal is to put a lid on a Möbius strip. The one we start with you will note is not just once twisted, but three times. I don’t know how essential this is to get an immersed projective plane at the end. I suppose it’s not, but makes things easier. Note that the strip has a single boundary curve, as expected.

The first two images show that Möbius strip, growing slowly. Below the first crucial step has happened: The growing strip has created a triple point, and intersection like that of three planes. But there still is only one boundary curve…

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We keep growing

 

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and growing:

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Another critical event: The boundary curve emerges completely into free air, i.e. doesn’t pierce through the surface anymore. Now it’s easy to close the lid:

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This Year in Marienbad

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Alain Resnais’ film L’Année dernière à Marienbad is generally praised as visually breathtaking and intellectually incomprehensible. Since this year, this film might also be called visionary.

A game is being played multiple times and one of the unnamed participants (called M in the script), states “Je peux perdre, mais je gagne toujours”. This sounds eerily familiar. And M does always win, making moves that don’t seem to follow any logic.

The similarities go much deeper. Both the actors as the viewers are not only left in doubt what is true or false (as in any good mystery), but also about what is real and unreal. The film takes place in a state of mind that has been dubbed hypernormality, a concept that Adam Curtis is using in his brilliant recent documentary HyperNormalisation to explain how our traditional perception of reality has been dismantled, with devastating consequences.

The game that is being played is called Nim, and it is at the center of the film for a reason. It is an impartial game, which means that both players have complete information (no hidden cards) and the same moves available (no black and white pieces owned by the players). Impartial games also must end with one player winning and the other player losing. This means in particular that either the first or the second player must have a strategy, proving M almost a lier, because he cannot have a strategy both as first and second player. He is, however, not claiming that he can always win, just that he does always win, thereby claiming access to a powers beyond those of reason.

Let’s have a closer look at Nim. It is played with a several heaps of tokens (matches in the film). At each turn, the player is allowed to take any positive number of tokens from a single pile. The player who takes the last token wins.

The simplest case is that of a single pile: The first player will win by taking the entire pile.

The second simplest case is that of two piles. Here, symmetry plays a fundamental role. If both piles have the same size, the player must necessarily take away from one pile, thus leaving two piles of different size. On the other hand, if the piles have different sizes, the player can take away tokens from the larger pile to make them equal.
This proves that there is a simple winning strategy that consists of making both piles equal in size.

We can visualize this using coordinates in the first quadrant: A game position with pile sizes x and y determines a square at coordinates (x,y).

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The olive green squares mark the positions where both heaps have the same size. To move means to decrease either the x or the y coordinate. We can clearly see that we can move from any white square to an olive square (winning move), and that we are forced to move from an olive square to a white square.

This is all very simple. However, as soon as the game is played with at least three heaps (the film uses four), things get much more complicated. Let’s see how the space of positions looks like. We can again use the first octant of space to indicate heap sizes x, y, z of three heaps by a little box at the point with coordinates (x,y,z). Below you see the boxes that indicate the losing positions for heap sizes 0 or 1 (left image) and heap sizes up to 3 (right image). A move again decreases precisely one of the three coordinates. Convince yourself that from one of the reddish boxes you have to move to a non-box, while from a non-box you can always move to a reddish box.

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You can also see that you get from the left image to the right image by substituting a box by the entire left image. This persists, and what emerges with increasing heap sizes is a fractal called the Sierpinski Pyramid.

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It is the full intention that this looks chaotic and complicated, because this is what a hypernormalised mind perceives. But behind this apparent chaos, there is a simple rule, except that its simplicity is not intuitively useful.

A position (x,y,z) is a losing position (and hence marked by a cube) precisely when the either-or sum of the binary representations of x, y, and z are zero. For instance, if the pile sizes are 1, 4, and 7, these decimal numbers have binary representation 001, 100, and 111. We obtain their either-or sum by adding these numbers in the binary system without carry, this gives 010. Because this is not 000, we are in winning position. The winning move takes 2 token from the third pile, changing its binary representation to 101.

This is computational very simple (and works for any number of piles), but there is no apparent way to make this intuitive. We humans do not feel that we are in a losing position in Nim. In this sense Nim becomes a perfect symbol for a world that appears detached from common sense, but can be controlled by algorithms.