Old And New (Summer in Berlin III)

There are many good places to contemplate the clashes between old and new in Berlin, and one of them is the area along the Spree near the U-Bahn station Schlesisches Tor. This is where the world ended for people living in West-Berlin while the city was divided. Now one can walk across the bridges and admire the construction circus on both sides.

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Herbert George Wells might have thought that his phantasies have come true. When they are done with all this, will it looks like this?

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And will we get more playful little sculptures like the Molecule Man by Jonathan Borofsky?

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There is some obvious resistance. It feels like the perfection of a finished building is stifling the creativity.
Who wouldn’t want to defend the octagonal brick building below?

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Do we really want to lose all this?

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My taste is more for blending old and new and let them coexist.

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Magnetism

Magnetism is played by two players on a strip of squares, who take turns placing + and – tokens onto the strip. The only rule is that no two tokens with the same parity can be placed next to each other. For instance, there are three legal moves in the following position:

Legal

The player who moves last, wins. This makes Magnetism an impartial game, so that each position is equivalent to a Nim-pile. It turns out that Magnetism is very simple.
First we notice that any position is the sum of simpler positions that have tokens just at the end of a strip. (A sum of games is played by first choosing a game summand, and then making a move in that summand).

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Therefore we will know everything about Magentism if we can determine the size of the Nim-piles (the “nimbers”) of the 9 elementary positions:

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Things get even simpler. Because of the symmetry of things, there are only four truly different boards to consider.

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Let’s denote the nimbers of a board of n empty squares (thus not counting the tokens at the end when present) by G(n), G+(n), G++(n), and G+-(n).

n 0 1 2 3 4 5 6
G(n) 0 1 0 1 0 1 0
G+(n) 0 1 2 3 4 5 6
G++(n) 1 1 1 1 1 1
G+-(n) 0 0 0 0 0 0 0

Now you can win in a position with a positive nimber by moving to a position with zero nimber. For instance, on a board with a single + at one end, one possible winning move is to put a – at the other end.

The Door Was Open (Summer in Berlin II)

I like architecture, or, to be precise, certain states of buildings. Ruins are fascinating, but even more so construction sites. Both are usually off limits (as are the corresponding states of human affairs, death and conception, unless you are involved one way or the other). So I am often forced to trespass a little.

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In this case, as you can see, the door was open, and I just couldn’t resist.

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Views like the one above make it instantly clear that we are not on a generic construction site. Somebody with taste has been designing this, and whoever is doing the construction work, is doing an excellent job by creating crystal clear previews of what’s to come.

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Wondrous tools are on display too, just for me. I can only guess their purpose by looking at the ornamented concrete slabs. Everything is purposeful, even the occasional leftover tile.

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What fascinates be most at places like these is the tension between the clarity of the present and the vagueness of an undefined future.

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Curved is also Beautiful

Among the many helicoids with handles, the translation invariant genus one helicoid is by far the simplest. It was first constructed by David Hoffman, Hermann Karcher, and Fusheng Wei. You can learn almost everything about it from a single image.

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The right hand side is a portion of the actual minimal surface, which extends by rotations about its horizontal and vertical lines to the complete surface.

Singlyperiodichelicoid

The quotient of this surface by its vertical translation is a torus, and the presence of the two straight symmetry lines hint that this is a rhombic torus, which you see outlined black in the left top left image. Its two diagonals become the two straight lines of the surface. The trick is to see the surface patch to the top right as the image of the colorful rectangle on the top left. The top left and bottom right corners of that rectangle are bent together so that they touch, the horizontal edges align as the horizontal line, and the vertical edges align as the vertical line of the surface.

The two semicircular arcs become the half turns of the two helicoidal arcs, this allows to truncate the surface image nicely. The mesh lines of the colored rectangle are, incidentally, obtained by conformal mapping a rectangle to itself in a quirky way:

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Here, the vertical edges of the left rectangle are mapped to the two semicircles, and the horizontal edges to everything else. The “extra” vertical lines are included so that we hit all vertices of the right rectangle by a parameter line.

So that is all very easy. The tricky part is make the right choices in order that the the two opposite corners of our parameter rectangle really meet. The horizontal alignment is achieved by using as a rhombic torus the funny 70.7083 degree rhombus we discussed last time. If you choose another rhombus, the two verical line segments will not match up.

To guarantee also a vertical alignment of the two corners, one needs to choose the location of the two points E1 and E2. To do this, one constructs a meromorphic 1-form on the torus which has simple poles at E1 and E2 and two zeroes at V1 and V2 (whose location depend on E1 and E2 by Abel’s theorem). The integral will map our colored rectangle to a slit domain consisting of two merged half strips. The ends of the half strip correspond to the two helicoidal ends of the surface.

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That the two slits line up in this picture is no coincidence. E1 and E2 have been chosen so that this happens (thanks to the intermediate value theorem). Tt is exactly what is needed to achieve the vertical alignment of the two corners.

The Jewish Museum (Summer in Berlin I)

Berlin has changed a lot since the wall came down in 1989. Most notably the constricted architecture from before finds its counterpoint in buildings that show a liberated sense of what can be done with space.

One of my favorites is the Libeskind addition to the Jewish Museum from 2001.

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You can only enter it underground and are confronted immediately with long and slanted corridors.

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I felt the natural way to photograph this is by slanting the camera as well. There is a lot of narrow vertical space,

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admitting just enough light so that we don’t feel claustrophobic.

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Then there are the Voids, most of them inaccessible, but present through views and gaps in our perception.

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We lose the distinction of being inside or outside, but we learn that is us who create the space around us.

Flat is Beautiful

Every flat 2-dimensional torus can be obtained by identifying opposite edges of a parallelogram.

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Each such torus has an involution that fixes four points, marked in four colors above. We can visualize the quotient as a tetrahedron with 180 degree angles at every corner by taking the triangle consisting of the lower left half of the parallelogram, and folding it together.

Tetra

So the space of all tori is nothing but the space of tetrahedra. Each such tetrahedron determines a unique point on the thrice punctured sphere. This can be seen by constructing the elliptic function on the torus determined by sending red to infinity, yellow to 0, and green to 1. The unique point is then the value of blue, and is called the modular invariant of the torus. To go backwards, take a point in the thrice punctured sphere and compute the quotient of elliptic integrals (using independent cycles)

Latex image 1

This complex number is the ratio of the two edges of the parallelogram that defines the torus.
This map is a Schwarz-Christoffel map: It maps the upper half plane to a circular triangle with all angles 0.
Restricting it to the upper half disk has as its image one half of such a triangle, namely

A 0 0 x

Let’s repeat all this starting again with a parallelogram, which now has been removed from the plane.

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Identifying again opposite edges defines a translation structure on a punctured torus that corresponds to a meromorphic 1-form with a double order zero at red (because the cone angle there is 6π), and a double order pole at infinity (yellow) (because the holomorphic 1-form dz has a double order pole at infinity). For a given modular invariant, we can determine the parallelogram to use using another Schwarz-Christoffel map

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which maps the upper half disk to a different circular polygon:

A 1 0 x

Curiously, we see that the ratio of the edge vectors of the parallelogram can now also lie in the lower half plane or even be real, in which case the parallelogram degenerates. For instance, we can construct a torus that corresponds to the quotient -1 by slitting the complex plane from -1 to 1, and identifying the top of the slit from -1 to 0 (resp. 0 to 1) with the bottom of the slit from 0 to 1 (resp. -1 to 0).

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This corresponds to an ordinary torus whose parallelogram is a rhombs with angle 70.7083 degrees. Next time we will see what this torus is good for.

The Price of Beauty (North Iceland XII)

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In this last post of my little series about North Iceland we return to Vesturdalur. Instead of revisiting the basalt cones at Hljóðaklettar, we hike another loop to Svínadalur.

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It begins with massive basalt formations. When in Hljóðaklettar they were well placed accents in the landscape, here one is overwhelmed to the extent that one doesn’t quite know where to look. Then the landscape unfolds.

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Who would want to live here?

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The little mounds are what is left of Svínadalur, a farm abandoned over 50 years ago. One can barely trace the contours of a handful of small houses.

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Who dares to build a home at a place like this where you have an open view of 50 km in every direction?

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And then, after having built that home, why would you leave? The trail continues towards the Jökulsá á Fjöllum and the views become more dramatic again.

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Maybe this place is too beautyful to bear it for too long.

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