Transcendence (Loxodromes I)

The art of map making took a giant leap in 1569, when Mercator created his first world map. Precise navigation had become an important problem. Seafarers not only had no GPS, they didn’t even have accurate clocks that would allow them to determine their longitude. One of the few reliable tools was, sadly, the compass. Therefore, a safe way to travel was to head in a direction of constant bearing, like say 20 degrees west of North.

Loxodrome

The problem, then, is: If you do that, where do you end up? On a sphere, the curves that make constant angle with the meridians, are called loxodromes. Mercator’s accomplishment was to find a map of the earth where all these loxodromes become straight lines. So, when you wanted to travel from A to B, you just had to find A and B on Mercator’s map, and measure the angle that the line through A and B makes with a longitude.

Cylinderprojection

This is equivalent to finding a map projection that preserves angles and where all longitudes are vertical lines. The Greeks (and maybe civilizations before) knew the cylindrical projection which is totally amazing because it preserves area, but it does not preserve angles.

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In fact, when you draw the loxodromes centered at a point on the equator on the rectangular map, you get curves that are clearly not straight (which is ok).

Loxorectangle

Nobody knows how Mercator came up with his map. It is believed that he just stretched the cylindrical projection so that the loxodromes became straight. But we don’t really know, and the reason is that the tools from calculus that are necessary to really construct this miraculous map were only developed centuries later.

Stereographic

There is another projection of the sphere, the stereographic projection, that was known to the Greeks. They at least knew that circles on the sphere would be mapped to circles or lines in the plane. It also preserves angles, which the Greeks could have known, because it is rather elementary. Apparently the first written proof is due to Edmond Halley in 1695 (using calculus).

Bispiral

The stereographic projection maps the loxodromes to logarithmic spirals (up above we use the loxodromes that connect west pole with east pole, for prettiness and later use). While the Greeks did study a few transcendental curves, the logarithmic spiral is first discussed by René Descartes in 1638 (in precisely the context of finding curves that intersect radii at constant angles), and a little later by the Bernoulli brothers, with analysis emerging.

Exprectangle3 01

Therefore it no surprise that the link between Mercator’s map and the stereographic projection is the (complex) exponential function (or logarithm). Today we know that it is angle preserving as one of the key features of complex analytic functions, but I don’t know who first realized this for the complex exponential function or the logarithm. Certainly Leonhard Euler deserves credit here. I doubt if it was earlier than the 18th century, even though the foundations were set by Mercator (possibly only by approximation) and Descartes centuries earlier. It is astonishing how long it takes to develop insights we now consider to be fundamental.

The Canadian Rockies

The prospect of the US elections this fall makes me (like many of my US friends) think about Canada.

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I went hiking there for two weeks with friends from California (sort of – one was from Britain, one from the US, and the third from Australia) in the summer of 1995. We hit two weeks of rain except for one day where it also hailed. Our planned week long backpacking trip needed major revisions. We tried to do overnighters on the trail, but it is not much fun to spend long nights in a tent while it rains all night (and morning).

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After that, we went for day hikes in the area during brief respites. Whenever the rain stopped, I got my camera out.

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At night, we mostly car camped under a tarp and spent hours discussing the problem in what positions one can move a given rectangular tarp by tying it to four given trees with ropes.

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It turned out that most of our arguments were wrong. Neither the weather nor the endless mathematical disputes had any negative impact on our friendship.

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A likely cause was the excellent Canadian wine.

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So maybe there is hope, after all. Next summer?

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Compass

Here is another game (like Demon) that utilizes the contemporary game mechanics of a game board that is being destructed during play.

To prepare, download the pdf file and print the five pages on heavy card stock, using a color printer. Then cut out the squares. This should give you sixty playing cards like these:

Singlecard 01

They each use all of the five different colors, so there are 30 different cards, and you get each card twice, which is what you want.

As preparation, the cards are shuffled and laid out in a 6×6 square.
The players choose colors and take two checkers in their color. Then they take turns placing their checkers in the middle square of a card. No two checkers may be on the same card at the same time. Like so:

Board1 01

Now the players take turns moving one of their checkers. A checker can move north, east, south, or west on any unoccupied card whose middle square has the same color as the border of the card the checker is moving away through.
For instance, to determine where the left white checker can move, we pick a direction (say east), look at the color of the border of the square the checker is on in that direction (red), and locate all cards in that direction (east) whose middle square has that color (red).

The next image shows all possible moves for all four checkers. Note that the top white checker can move one but not three squares south, because the latter square is occupied.

Moves1 01

After moving a checker, the player collects the card the checker moved away from. If a player cannot move anymore, (s)he can either call the phase of the game over, or place one her/his collected cards on an empty square, in any orientation. Her/His move is then over, without moving a checker.

When the first phase of the game is over, the second phase begins. Here, the two players use the cards they have collected to create jewels: These are 2×2 squares where common edges have the same color:

Jewel 01

The jewels score differently, depending on the number of different colors in their shared borders.
If all are colors of the common borders are different, the jewel is worth 1 point,
if three different colors occur as shared borders, the jewel is worth 2 points, and
if three different colors occur as shared borders, the jewel is worth 3 points.

So the jewel above scores 2 points. The player with the most points wins.

The Underneath

The Underneath is the title of a book by Kathi Appelt that I and my daughter really enjoyed reading. It has, however, nothing to do with this post but its compelling title.

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What would the world look like if we were as little as we righteously should be: Suppose we were bug-sized, waiting for food or to be eaten in a forest of may apples. What would we know of the larger world?

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Above us the strange smelling flowers, and below the decaying leaves from last fall? Tough choice.

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Or, imposing and obviously hungry giants. Would they eat us, too?

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Maybe there is a protective cave behind that tall waterfall?

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One of the good things of being little is that even small rocks like these become impossible to lift.


Welchen der Steine du hebst …

Periodic k-Noids (Minimal Surfaces in the Wild II)

The k-Noids that Shireen built last winter will keep roaming the Swiss landscape, from May to August in Wülflingen. Maybe it is time to corral them. My suggestion is to build fences of catenoids. The most classical one looks like this:

Fence1

It is, technically, a translation invariant minimal surface that has two ends and genus 1 in the quotient. A simple generalization and an even simpler 90 degree rotation leeds to towers with catenoidal openings.

Fence2

If that isn’t safe enough, you can have them with double walls (i.e. with genus two in the quotient) like so

Fence3

or so:

Fence4

All these examples have many ends in the quotient. The surprise is that there also is the elusive Uninoid which only has one catenoidal end in the quotient, namely by a 180 degree screw motion:

Fence5

Here things get tricky. Michelle Hackman has found more complicated versions of this in her thesis. Here is a Uninoid that is invariant under a screw motion with quarter turn.

Fence6

Trillium Luteum

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The Trillium luteum is a yellow variant of the red Trillium Sessile. Like the sessile, the flower sits right on top of the three symmetric bracts at the end of the scape.

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These pictures were taken in 2005 and 2006 in McCormick Creek’s State Park. In April one can find there mostly the Trillium Sessile, the Snow Trillium and the drooping trillium.

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Back then, I had just started taking pictures of the local wildflowers, and took pictures of everything that didn’t appear too common. Sadly enough, I have never seen a yellow trillium again in Indiana.

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A Sense of Space

My first encounter with Virtual Reality was in the 1980s, when text adventures became popular on the new affordable desktop computers.
We spent countless hours trying to figure out what to do with the pocket fluff in the text adventure version of Douglas Adams’ The Hitchhiker’s Guide to the Galaxy, made by Infocom.

Many of these games shamelessly exploited the limitations of their virtual realities: Because all interactions are verbal, there is always the possibility that what reads like a visual description of a place can in no way represent a real place. This gives plenty of opportunity for devious puzzles and mazes. A few years ago, I came up with my own little nightmarish maze, called the Un-Maze. You can play it here in a web browser. It is very bare bones, but it will tell you when you have found the exit.

Maze0 01

The rest of this post will explain this puzzle, so don’t read on yet of you like a challenge. Let’s make it simple, let’s imagine a maze where every room has only two exits, called left and right. We might think of this maze as an infinite sequence of rooms. If it happens so that all the rooms look alike, and we have no means of altering the appearance of a room, we could also be just in a single room whose right exit leads through a twisting passage to the right exit.

Maze1 01

My Un-Maze up above lets you decorate the rooms a little bit, because you can pick up and drop three different pebbles, and when you type “look”, the game will tell you whether there is a pebble in your current room. A basic unjustified assumption we make about such mazes is that when we exit a room to the right, we should be able to get back to that room by exiting the next room to the left. Many mazes in text adventures warn you about this, by telling you that you are entering a long winding passage.

The simple idea behind the Un-Maze is that your location in space is solely determined by your previous actions. For instance, if you decided to walk left-right-right, then you are in the room left-right-right.

Unmaze0 01

This is yet another model of a strange universe where in every room we can only move left or right. This infinite tree assumes that we have unlimited memory. What happens if we can only remember our previous three actions? Our universe would look like this:

Unmaze1 01

We change the name of a room by forgetting its first letter and appending the first letter of the action we took to get there (Left or Right) to its end.

If, for instance, you knew that the exit to the maze was at room LLL, you could reach the exit from any room my going three times left. This is still a maze were all rooms look exactly the same. To change this, we can remove some exits. In the following maze, we have removed the possibility to turn right from some rooms, and now it takes five turns to get from LLL to RRR:

Unmaze2 01

The text adventure maze features four directions and the rooms are given by the memory of the last two turns. You found the exit if you manage to first go east and then north. Good luck.