Jerome, oh why you treat me so cold? (Arizona IV)

Another example how to deal with days past can be admired in Jerome, an old miner’s town. Today’s population is a mere 5% of what it used to be 100 years ago when copper was plenty. So, where did all the people go, and what happened to the houses?

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The ones that appear not to be abandoned cultivate lucrative traditions

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or celebrate an unprofitable business by placing it at a location that makes it particularly difficult. How would you arrange the book shelves?

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For the curious tourist, accommodations are plenty,

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and transportation is traditional.

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Then there are the true and timeless inhabitants, always slightly annoyed.

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Spring Cleaning I

Spring Cleaning is played on a rectangular array of randomly placed dirt pieces. A sweep consists of removing a single row or column of consecutive dirt pieces.

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Above are some example of legal sweeps, and below are illegal sweeps.

Sweep illegal

This is a game for two players, who take turns by doing exactly one sweep. The player who sweeps the last time is the winner. This is an impartial game which means that each position is equivalent to a single game of Nim. This is usually bad news, because playing Nim well requires us to perform exclusive or additions of binary numbers in our head, for which our brains are not (yet) well equipped.

The good news here is that many simple positions are equivalent to very small Nim piles, meaning that computations are easy. I will explain this using an example. No proofs (even though they are easy, too).

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It’s your turn to find a winning move in the position above. You know (because I promise) that rectangles completely filled with dirt pieces are easy positions, so you will look for moves that separate the dirt pieces into such rectangles. Here is such a move:

Sweep win1 sol

After that, we are left with four separate rectangles, all completely dirty. This means that this game is equivalent to a game of Nim with four Nim piles. The question is what the pile sizes are. The answer is simple: Any rectangle both of whose dimensions are odd corresponds to a Nim pile of size 1, if both dimensions are even, the Nim pile is empty (size 0), and otherwise, the Nim pile has size 2. In our example, we have a 1×1 rectangle, a 1×3 rectangle, and two 1×2 rectangles. They correspond to Nim piles of sizes 1, 1, 2, and 2. The exclusive or sum of these numbers is 0. This is what we want, because it means that after this move, the game is equivalent to an empty Nim pile. From now on it’s easy. Suppose that our opponent performs a vertical swipe on the 1×3 rectangle. What do we do to return the game to Nim-value 0?

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We can sweep away any of the isolated dirt pieces: From then on, the game is symmetrical and we can win easily without any Nim-theory. And we better leave the two 1×2 rectangles untouched. Suppose we remove one of them completely. Then we are left with three 1×1 rectangles and a single 1×2 rectangle, which exclusive or sums up to a Nim pile of size 3, in which case our opponent can win.

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The winning move would be to reduce the remaining 1×2 rectangle to a 1×1 rectangle with a horizontal sweep.

So, if we know how to deal with Nim positions that consist of Nim piles of sizes 1 and 2, we will be able to win Spring Cleaning by dissecting a given position eventually into rectangles.

Recovery (DePauw Nature Park I)

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The main attraction of the DePauw Nature Park is its limestone quarry. It is a vast and eery place. After DePauw University took possession of the area, they removed buildings and other environmental damage, and allowed nature to recover.

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So one can see the effect of two different forces at once: The enormous momentous force of machines that cut through all that limestone, and the much slower force of plants that take roots again.

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This appears to be simple. We know the constituents: water, rock, grass, tree. But then there is weird stuff I don’t know. I will pretend it is harmless.

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Maybe Mars looked like this, too, millions of years ago.

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Playing with circular images

After successfully transforming rectangular images into circular ones it is time to do something with them. We have seen already that the one can deform them by shifting one point somewhere else. This is very much like rotating a globe.

But besides these angle preserving symmetries of the disk there are other maps from the disk to itself that also preserve angles but are not anymore 1:1. These are the Blaschke products, written in complex notation as follows.

Product 01

Let’s look at a simple example with just two factors, and choose the a-parameters to be 1/2 and -1/2. Then B(z) maps the double spiderweb on the right to the standard spiderweb on the left:

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In other words, by taking preimages (or better, by using B(z) to pull back an image…), we can create multiple copies of a circular image within a circular image. The a-parameters designate the locations of the “centers” of the multiple spiderwebs where the strands converge.


For instance, above is a circular image of a Spring wild flower, and to the left its 3-fold mutation. Below are 5-fold mutations with two different choices for the a-parameter.


These images resemble kaleidoscopes, but are improved, because the copies of the original image fit together more evenly (smoothly, and not just by reflection). One can also make the result less symmetrical by choosing the a-parameters less symmetrical. Below the copies of the ferns are places at 120 degree angles but differently far out,


and here we have a large copy of the original budding trillium at the bottom with two smaller copies to the left and right.


Now I need to find somebody who writes an app that implements all this…

Down (Arizona IV)

When you look down in Indiana, you see either mud or decaying leaves. This is of course exaggerating it, but the contrast to Arizona is so stark that I ended up taking a considerable number of pictures by just pointing the camera downwards.

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It of course always depends on where you are and what you do. I don’t envy the brave NASA scientists who have been staring for decades at red desert rocks from Mars. What will the first plant on Mars look like?

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Then there are the forests, smelling of pine and juniper.

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The proximity of decay and growth shows how fragile is what we have,

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and how much it depends on water.

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A Room With Two Views (Five Squares IV)

Today we look at tilings that utilize just the four other squares. The first step in classifying these is again a simplification, making the split corner squares uniformly green. This leaves us with two tiles:


Ignoring the pink triangles for the moemnt, we recognize the problem we solved last time: The green squares need to occur in shifted rows or columns, like in the example below. Here we have four rows of green squares. Rows 1 and 2 are shifted, as are rows 2 and 3, but rows 3 and 4 are alined.


To add the pink triangles, note that two pink triangles fit together to a pink diamond, and each grid cell needs to have one of those, but we can only use those edges that are not already adorned with a green square. This leaved us with the following possibilities: If two consecutive rows of squares are aligned, we have place two diamonds in the square space between four squares, and we can do this horizontally or vertically. This can be done independently of neighboring squares, as shown between the two bottom rows below.


If the rows are shifted, we also have two possibilities to place the diamonds, but each choice affects the entire row, again as show above in the top rows.

Finally, we need to undo the merging of the orange and blue triangles into green squares, and we can do so by splitting each square either way and independently.


Below is an example how teh corresponding polyhedral surfaces will look like. The horizontal squares correspond to the green squares of the tiling. They are the floors and ceilings of rooms that have two opposing walls and two openings. I start seeing applications to randomly generated levels of video games here…


Oscillograms (Arizona III)

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About an hour car drive away from the desert landscape of the Petrified Forest in Arizona, one finds oneself in the large National Forests of Arizona. Change can happen quickly.

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For a little while, melt water from winter snow leaves scenic lakes where tall pines try to protect the smaller birches in early morning light.

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Summer draughts and quickly progressing privatization threaten all this.

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After sunset, when the few humans have retreated into their safe houses and the winds have subsided, the landscape becomes very quiet. The perfect reflections of the resting trees look like oscillograms of unheard cries.

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Change happens quickly.

Squaring the Circle

Squaring the circle is easy, you just need to know what you want to do. My personal favorite method is to use elliptic functions defined on rectangular tori to map rectangles to disks, as shown below for a square. These maps don’t preserve area (which is what the Greeks had wanted), but they preserve angles.


I had some leftover architecture images from Columbus and wanted to see how they look when made circular. Here, for instance, is the AT&T building

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and this is a circular version:


There are three degrees of freedom one can play with (the dimension of the automorphism group of the hyperbolic plane), which means that one can squeeze parts of the image towards the boundary cirle. Here are two other versions of the same image.


Another favorite of mine is the atrium of the Cummins office building with its wonderfully intricate play with straight lines and black and white.

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Now we only have to find architects and builders who create buildings that have these curves in reality.