Dissections and Area

Whenever need to explain what Mathematics is about, one of my favorite examples is the concept of area. The existence of an elementary notion of area hinges on the fact that any two Euclidean polygons have the same area if and only if they are scissor congruent, meaning that they can be cut into congruent pieces using straight cuts. To see this, it suffices to show that any polygon can be dissected into a square.

The example above shows how to dissect a well-proportioned rectangle into a square. Here, well-proportioned means that the rectangle is not more than twice as tall than wide. If a rectangle is not well-proportioned, a few cuts parallel to the edges will make it so. Thus any two rectangles of the same area can be dissected int each other. We will use this later.

Next we show that any polygon can be dissected into triangles. By induction, it suffices to find a secant inside the polygon. To find this secant, pretend that the polygon is actually the floor plan of a room, and we are standing at one vertex V . The two adjacent walls lead to two vertices A and B which we can see. If we can see yet another vertex W from our position, we have found our secant VW. If we can’t see another vertex, nothing obstructs our view in the triangular region formed by A, B and V , and thus A and B can be joined by a secant.

As a further simplification, we cut all triangles into two pieces along one of their heights so that all triangles become right triangles.

Now we have a collection of right triangles, which will need to be dissected into a single square.

To do so, we dissect each right triangle into a rectangle. This can be done as shown above by dissecting the triangle into two pieces along a segment parallel to one of the legs and dividing the other leg into equal parts.

This leaves us with a collection of rectangles that most likely have different dimensions. So we dissect them into new rectangles that all have all height 1, using the example at the beginning.

Then, the new rectangles can be lined up edge to edge along their sides of length 1 to form one very wide rectangle that finally can be dissected into a square.

As this was nice and easy, here a challenge: In our dissections, we were allowed to translate and rotate the pieces arbitrarily. What about if we forbid rotations? Can you dissect an equilateral triangle into finitely many pieces and translate them so that the result is the same triangle upside down? Or, can you cut a square, translate the pieces, and thereby achieve a 45 degree rotation of the square?

First Times (California ’93)

My little series with  pictures from 25 years ago continues with my first hiking trip in California. The idea was to drive up to the trail head of White Mountain Peak, and hike the dirt road to the peak.

We drove through Yosemite at night (which I hadn’t seen before) and camped at my first hot spring in Owen’s Valley. Soaking in warm water while around you everything freezes and the sky is full of shooting stars convinced me that this had been a good idea. We made it past the Bristlecone Pine Trees, but the car didn’t make it to the trail head (my first car break down).

We didn’t give up though but continued on foot. The landscape up there (above 10,000 feet) is high elevation desert.

After two hours or so we reached the observatory and the actual trailhead. Hiking appears very easy: You just follow the dirt road.

What is not so easy, however, is the high elevation. Two of use got altitude sickness, including myself (first time!). That was interesting. It started off with gradually worsening headache.

After a while, my vision got blurry, and me and the other victim turned back to the observatory. While we waited for the two others to return from the summit, we chatted with the friendly personnel. By 10pm, the two other hikers had returned, and we were lucky to hitch a ride on a pickup truck back to our car.

The next morning we stopped for my first visit at Mono Lake.

President (Impeachment Games I)

Time for another little game. It’s called President, and it is also about democracy and taxes. Good for 4-12 players.

Material:

• Paper money (any currency will do)
• sets of six tokens in as many different colors as there are players,
• a tax board for each player, i.e. a sheet of paper with the numbers from 0 to 60 in a row,
• an extra counter to mark the current tax rate,
• 2-4 game figures representing political parties.

Setup:

Each player gets \$100 in small bills, picks a color and gets three tokens of that color,
sets the tax rate counter on her or his tax board to \$10.
The political parties are placed in the center of the board.

The game proceeds in years. Each year consists of an election phase and and ruling phase.

Before the game begins, the players decide for how many years they want to play.

Voting:

First, all players pay their taxes.
For the first year, these are \$10, and each player places them into the center of the table.
If in a later round a player cannot pay the taxes, that’s ok. The poor are tax exempt. But see the variation below.

Next, each player votes for political parties by placing their tokens next to the party figures. Votes can be split, and not all votes need to be used. In the first round, a player is selected randomly to begin the voting, and then it continues clockwise.

After all votes have been cast, the winning party is determined by counting all votes. If more than one party  has the same highest number of votes, the winning party is determined randomly. You can place the tied parties in a bag and draw one, for instance.

The player who cast the most votes for the winning party is declared president. If there are more than one player with the same highest number of votes, the president is determined randomly among all players with the highest number of votes for the leading party.

All used voting tokens are returned to the players, which ends the voting phase.

Variation: If a player cannot pay taxes, he/she loses one voting counter.

Ruling:

In the ruling phase, the elected president must make a few decisions:

• Adjust taxes: The president can adjust how much taxes each player pays. He or she does so by moving the tax counter on each player’s tax board up or down by at most \$2. The taxes cannot exceed \$60 and must be at least \$10. The total amount of taxes paid must remain equal to \$10 times the number of players. This means that if the president lowers taxes somewhere, he/she must increase them somewhere else.
• Adjust the number of votes: For each player, the president can increase or decrease the number of voting tokens the player has by at most one. The number of voting tokens cannot exceed 6, and must be at least 1.
• Distribute tax income: Finally, the president distributes all of this year’s tax income to all players as he or she desires, including him or herself.

The game ends when the number of years the players agreed on has passed. Then the plaeyrs vote on one of the following three winning conditions:

• The richest player wins
• The player who was most often president wins
• The player who has the most voting tokens wins

That was a lot of text. I like to invent games, and I like to watch how people play games. So I programmed a little simulation to see how this game would perform while tweaking the parameters. All players in my simulation behave opportunistic. They begin with equal preferences for the political parties. When somebody’s income increases, they start favoring the ruling party in the next vote. The president uses his/her power to give more votes and more money to those who voted for his/her party.

In contrast to humans, the computer was willing to play for an extended period of time. I was expecting that the game would quickly stabilize to a single rich dictator with the rest of the population living in poverty. The pictures above show the wealth/time graph of a 4-person game with just two parties. While presidents often rule for long periods of time (50,000 years for the blue president…), the situations is all but stable. That it can take so long is only because the underlings in my simulation do not cooperate. Below are mere 5,000 years with eight players and four parties. I have run several such simulations, and it appears that change happens more often when there are more parties involved.

No Lack of Color

One of the things one can do in early Spring in California is to go to Lassen Volcanic Park when the crowds are not there yet but the snow is gone so far that one can actually get into the park. This is place of stark contrast. There is a steep and rocky hike up Mount Lassen which offers nice views, for instance to lonely Mount Shasta.

At lower evaluations you can hike thorough lush forests to what I consider the most stunning part of the park, the Painted Dunes and Cinder Cone area.

The landscape transforms within minutes into something that borders on abstract art.

This is a pretty remote area of the park, in particular when the access roads are still closed in winter. Mayhem can happen quickly.

These pictures are now 25 years old. At some point I will need to check out how it looks today.

Deceptive Similarities

This story begins in 1988 with the first examples of doubly periodic embedded minimal surfaces where the top and bottom ends are parallel and asymptotic to vertical half planes. They were found by Karcher and Meeks-Rosenberg in two independent papers and look like these:

I like it how confusingly similar they are. The main distinction is that that the one on the left has horizontal straight lines on it about which you can rotate the surface unto itself, while the one on the right has a reflectional plane of symmetry instead.

These surfaces actually come in a 3-parameter family; what you see above are the most symmetric cases. The translational periods are horizontal, and there are vertical straight lines. If you divide by the translations, you get a torus as a quotient surface. Remarkably, this 3-dimensional family is all there is for genus 1, by work of Pérez, Rodriguez, and Traizet from 2005. In particular this means that the two surfaces above can be deformed into each other through minimal surfaces. This is not too hard to see.

Things get more interesting already at genus 2: At 1992, Wei found a 1-parameter family of examples of genus 2.
A specimen is below to the left.

A variation of it was constructed by Rossman, Thayer and Wohlgemuth in 2000 (above to he right). Again they look dazzlingly similar. I suspect that they cannot even be deformed into each other through embedded minimal surfaces, but I have no idea how to prove this.

Even better, Connor has numerically found another genus 2 example that looks significantly different from the ones above. One of the holes has is not symmetrically positioned anymore, and there is no way to get it back there…

Showing the existence of these surfaces would be good, and even better would be to find a way to distinguish it from the others.

Patience

No, winter isn’t over. While we are waiting impatiently another one or two months for the first wild flowers to come out, Nature itself appears to be very patient

These sycamore fruits have been hanging there all winter.

Around them is proof that there has been a future.

This is not Waiting for Godot. Instead, this is comfortable trust: L’enfer, c’est les autres.

Cutting Corners

The two psychedelic designs up above arise from their simplistic ancestors we looked at last time by cutting off corners. These are still two conformal annuli that also satisfy a slightly complicated condition on the lengths of their edges, which makes them responsible for a variation of the Diamond surface:

If you cut either of the psychedelic shapes into quarters, using a vertical and a horizontal cut, you get four right angled octagons, with some right angles being exterior angles. Similarly, the marked symmetry lines on the surface up above cut the surface into eight right angled curved octagons, that correspond to the psychedelic octagons via a conformal and harmonic map.

There is a 1-parameter family of such critters. Above and below are larger portions of extreme cases that also show how the surface repeats.

You can see in the image above pieces of the doubly periodic Karcher-Scherk surface reappearing. No surprise, its psychedelic polygons also arise by cutting corners in the polygons corresponding to the original Scherk surface.

Everything simple reappears.

Winter Was Hard

It’s time to say goodvye to winter for this year, and a good way to do so is with some pictures I took with Lensbaby’s Velvet 56.

The completely frozen creek offered easy walking (wearing cleats) to familiar places. The imminent thawing will dislodge and transform. Some things age quickly.

One can hope.

The title of this post refers to a piece for String Quartet by Aulis Sallinen, and a CD with the same name by the Kronos Quartet from 1988. That’s 30 years ago. Some things age slowly.