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).

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.

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.

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.

## The Zone

In his film Stalker, Andrei Tarkovsky transforms the mysterious zones from Arkady and Boris Strugatzky’s book Roadside Picnic into a spiritual personal experience for the visitor.

I like that concept of a place that is off the map where we can go and dream or contemplate.

Fortunately, it is possible in Indiana’s slightly boring landscape to just step off the path and end up in one’s own little zone.

Granted, there will be no alien artifacts to collect, and no wishes fulfilled. But that was never the true purpose of Tarkovsky’s zone. Instead, a zone allows undisturbed introspection.

Sadly, these undisturbed space become harder to find, and maybe we have to move the zones into a virtual space. When the space limit on this blog runs out, I might call my next blog The Zone.

## Pillowminoes (From the Pillowbook VI)

The admission of an abundance of pillows with straight edges to the zoo raises the question whether these new citizens are any good. We have employed the ones with two straight edges to form arrows and combine to Hamiltonian circuits. Today we will look at those eight pillows with a single straight edge (let’s call them the singles)

Can we use them to tile a curvy 7×7 square, say? The answer is clearly no, because the singles have to hide their straight edges by combining in pairs to pillowminoes. This means we can only tile curvy shapes with an even number of singles. Here, for instance, is a simple solution that shows how to tile a 7×7 curvy square with a gap at the center. It also shows how to tile this shape with a single pillowmino.

This looks too easy? Can we also tile the same curvy 7×7 square so that the missing square at the center has four straight edges?

A little trial and error shows that this is not possible, but we would like to have a reason for this. We need for singles to neighbor the missing square at the center with their straight edges. I have indicated their position by a slightly darker shade of green. Thus the remaining lighter green squares will be entirely curvy, so needs to be tiled with singles that have combined into pillowminoes. That, however, is impossible: Color the squares alternatingly yellow and pink, as in the solution above. Each pillowmino will cover a pink and a yellow square, but the light green shape that needs to be tiled consists 24 pink and 20 yellow squares. This argument also shows that the missing square needs to have all edges curved.

What else can we do with the pillowminoes? There are 36 of them, and not all of them tile by themselves. If we want to tile the curvy 7×7 square with a circular gap in the middle, we will also need to balance the convex and concave edges, as explained earlier.

Here are the ten balanced pillowminoes:

Surprisingly, only the top left will tile the 7×7 (or any larger) curvy square with a central gap.
Below is an example that tiles with four individually unbalanced but centrally symmetric pillowminoes.

More to follow!

## Dendrochronology

When helplessly confronted with historic events, we can be little more than a witness. But we should not underestimate this task: Being a fair witness is both difficult and necessary. This is a skill that should be taught in school.

I have always admired the trees as such witnesses, and maybe the tree can become the name of a pattern that describes the functions of a witness.

The tree images of this are also a personal memory, because they were taken near the event horizon when my own personal history becomes imageless because I don’t have photos from earlier years.

So all these images are from the early 80s, taken near Bonn. I had moved there from (West) Berlin and was beginning to learn that one can spend one’s free time exploring the surroundings.

The shivering tree below is not out of focus. It is a double exposure, with the second image being a long time exposure to motion blur the leaves in the wind.

## 2-3-6

I like it when apparently simple things evolve all by themselves into complex objects. Like watching cactus seeds grow into cacti. That was a distraction, but I do like it. Below is a left over piece of mathematics that would have fit nicely into a paper I wrote with Shoichi about triply periodic minimal surfaces.

It is, evidently, quite complicated. To unravel it, here is a smaller portion of it, its seed, so to speek.

That is a minimal surface inside a prism over a 2-3-6 triangle (which has a right angle, a 30 degree angle, and a 60 degree angle).
The curves in the vertical faces of the prism are symmetry curves of the surface, and reflecting at these faces of the prism extends the surface. The two curves in the bottom and top face of the prism are not symmetry curves, but when you place two prisms on top of each other (by translation), the curves will fit. The pattern the curves make on the prism determines the surface almost completely, there is just one degree of freedom. Here is another, equally pretty, version, using a different parameter.

Another way to seed these surfaces is through conformal geometry. Below is the conformal image of a circular annulus onto a polygonal annulus bounded by two nested 2-3-6 triangles. The parameter lines are images of radii and concentric circles, respectively. This map is the main ingredient in the Weierstrass representation of all these surfaces. Simple, isn’t it?

## Preparing for the Future

For a democracy to function properly, two things are essential: Firstly, the voting population must to some extent be able to identify with the entire population, meaning to be able to look beyond their own belly, and rather to use common sense. Secondly, at all times it must be the clearly stated intent of all candidates to protect the interests of all minorities.

Sadly, neither of these prerequisites were met in the US election. An excellent way to practice them is by playing games. It teaches you that you have to play by the rules, and that decisions based on rational thinking are usually better than those based on your guts.

Below is a slight modification of a game I made up for school age children a while ago, allowing us to experience some aspects of daily medieval life. Let’s prepare for the future.

# Material

• The game board shows a track like in monopoly, ruled by lords and disgraced lords.
• A deck of some 50 small cards showing loaves of bread. There should be cards showing 1 and 5 loaves. Make you own.
• Standard die. If you want the game to be more realistic, get a loaded die.
• Play figures for each player, all white. You know who you are.

# Preparation

All players choose a play figure and put it onto Harvest. They each roll the die twice to determine the harvest: The total number of eyes is the number of bread each player receives. The player with the fewest breads begins. From then on, the game proceeds clockwise.

# Turns

The player rolls the die and moves her figure clockwise as many fields forward as the die shows. Depending on the field, the player takes the associated action.

# Property field

If the player lands on a property field, she can decide to seek employment or to steal something.
After this decision is made, she rolls the die.

## Employment

The reward depends on the number of eyes the die shows:

1. you obtain work and are paid with one bread.
2. you obtain work and are paid with two breads.
3. you are allowed to work without payment.
4. you stay for the night and pay one bread.
5. you are robbed and lose two bread.
6. you are suspected of stealing. Move directly to ordeal. At your next turn you will be tried.

## Stealing

The success depends on the number of eyes the die shows:

1. you steal 1 bread and escape unrecognized.
2. you steal 2 bread and escape unrecognized.
3. you steal 3 bread and escape unrecognized.
4. you are sent to jail and stay there for one turn.
5. you are sent to jail and stay there for two turns.
6. you are sent to jail and stay there for three turns.

# Jail

If you land on jail, and the jail is empty, nothing else happens. If there are one or more prisoners in jail,
you free one of the prisoners of your choice but have to take her punishment instead. For instance, if a prisoner has to skip two rounds and you free her, she is moved to the white visitor spot in the prison while you are placed in the gray prison cell with number 2 in it. After you skipped two turns, you will continue.
When you are in jail already, each turn you move one step to the left to remind you of the number of turns still to skip.

# Ordeal

If you land on Ordeal, you are accused of stealing. To find out whether your are guilty, you roll a die. Your punishment depends on the outcome of the die:

2. Guilty: go immediately to Pilgrimage. Do not collect bread at Harvest.
3. Guilty: go immediately to Crusade.
4. Guilty: go immediately to Shaming.
5. Not guilty. You may continue on your next turn.
6. You accuse another player of your choice of witchcraft and go free on your next turn. The other player is put in jail and misses three turns.

# Shaming

If you land on or pass shaming, you may give any of the other players on Shaming one bread out of mercy. This player will then go free on her next turn.
If you have been sent to shaming, you have to wait there until a player passes you and gives you one bread. You then keep the bread and are allowed to continue on your next turn.

# Other Action Fields

These come with explanations:

• Church: Pay one bread for penance
• Broken Leg: Wait one round
• Dowry: Pay two bread for daughter
• Pilgrimage: Wait one round
• Treasure: You find gold worth five bread
• Harvest: If a player lands on or passes Harvest, she rolls a die and obtains the corresponding number of bread.

If a player is out of bread but has to pay, she goes to Shaming. There she waits until another player passes her and gives her one bread.

# End of Game

The game ends either after six rounds or when all players are stuck in Shaming.
In the first case, the player with the most bread wins. In the second case, all players lose.

## Points of Support

Three points make a triangle.

After witnessing the remnants of the American democracy in free fall during this year’s election, I found some peace in contemplating the stability that is achieved by three points
on a long walk through Turkey Run State Park’s maze of canyons.

Fall is almost over here, and the dead leaves and trees are awaiting the mercy of frost and snow.

There is beauty still in all of this, maybe because it cannot be made a target of hatred.

Let’s call this a prayer, if such thing still exists.

## Yellow

When Giacinto Scelsi went through a personal crisis, he spent hours listening to the sound of a single key on his piano. The haunting music he composed afterwards is my favorite music for the Fall.

Appropriately once in a while the leaves make a concerted effort to display just one color.

The yellow fall color is particularly difficult to catch, because the leaves fade too quickly when on the ground. This fragility makes yellow ideal to increase tension in an image.

I have difficulties pinpointing what I like about the image below. I keep returning to the pale yellow tree in the center that seems hover over the slanted lines of nearby rocks and trees,
ready to exert more pressure downwards, but not quite doing it yet.

Fragility is also part of Scelsi’s music. The first time I heard his music was at a concert in Cologne, and one piece was for solo guitar. Half way through the piece a string broke. The player just sat there for a minute, seemingly uncomprehending. When he recovered, he looked into the audience, went up to get a new string, and recommenced playing. This one minute of silence could very well have been part of the music.