Month: June 2017

Möðrudalur (North Iceland IV)

Möðrudalur is a small settlement in the northeast of Iceland. It used to lie along the ring road, but modern efficiency has bypassed it.

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Too bad, because that alternative to the ring road, called Möðrudalsleið, is much more exciting. There are dramatic views of the Herðubreið (which was, unfortunately, not yet accessible when we visited).

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In contrast to the ring road, it is just a simple dirt road, but not too bad.

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It climbs up the mountains and offers views of otherworldly beauty.

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Heinrich von Kleist, when confronted with the landscape paintings by Caspar David Friedrich, wrote famously that it felt like one’s eyelids had been cut away. This landscape feels like this, too.

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That this is so vast and still so simple creates confidence in its permanence. There is nothing to destroy here, and even if major earthquakes or volcanic eruptions take place, it will still look pretty much the same. Human souls should be like this.

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Hljóðaklettar (North Iceland III)

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Let’s continue last week’s post with more basalt structures. The place to be is Hljóðaklettar in the Jökulsárgljúfur National Park. To get there, you turn south on 862 a little west of Ásbyrgi. There are a couple of well marked trails in this area; today we follow the river north. The landscape is marked with distinctive humps

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that consist of clusters of basalt columns.

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How much time did it take to build all this? We humans are truly ridiculously short-lived.

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Then there are also the half-humps, like split giant geodes.

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The inside (i.e. the left side of the hump up above) shows more strange rock formations

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that up close seem to look at us with mild disdain. Rightly so.

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Aldeyjarfoss (North Iceland II)

Within two hours driving from Húsavik, there are plenty more or less easy to reach places of interest. One of them is the Goðafoss waterfall, which is visible right from the ring road. Nearby, but not quite so easy to reach is the Aldeyjarfoss.

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To get there, one follows 842 south (a dirt road, better than 844, which is an alternative). This turns after a few bumpy kilometers into F26. Most people drive their two wheel drive cars up to the parking lot.

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The waterfall itself is quite impressive, but its real beauty is due to the large basalt formations surrounding it.

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Next to it are some more contemplative smaller falls,

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and a short hike takes you to another large fall, the Hrafnabjargafoss.

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On the way, the rock formations on the river banks have the appearance of ancient friezes, telling stories about civilizations long forgotten.

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The complexity of this place made of rock and water is quite overwhelming.

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Drawing by Numbers

An easy attempt to make a model of color space is Philipp Otto Runge’s color sphere from 1810. The equator is colored by hue, brightness ranges from black at the north pole to white at the south pole.

Having such a color sphere suggests yet another way to visualize complex valued functions: Color a point z in the domain of the function by Runge’s color of the function value of that point, interpreted as a point on the Riemann sphere.

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For instance, up above is the coloring of the function f(z)=z^2 in the unit square. The white region at the center is caused by the zero, and that every color appears twice is of course a consequence of f being of degree 2. Let’s make it slowly more interesting. Here is the Möbius transformation f(z)=(z-1)/(z+1). Zero and poles are clearly visible.

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Locally, holomorphic functions are just as good or as bad as polynomials, so we shouldn’t expect anything more complicated to happen.

For me, the real excitement of complex analysis starts with essential singularities. There are the mind boggling theorems of Casorati-Weierstrass (images of neighborhoods of essential singularities are dense) and, much stronger and much harder to prove, Picard (images of neighborhoods of essential singularities miss at most two points).

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Above is the coloring of f(z)= e^(i/z) in a thin rectangle centered at 0. We do see every color occurring infinitely often (more or less), but the image is still rather tame. After all, the exponential function is the simplest transcendental function. Things get truly wild if we look at the boundary behavior of gap series like

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This series converges in the unit disk but cannot be homomorphically extended beyond.

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Unfortunately I don’t know enough about gap series to explain everything we can see here. Most puzzling are the circular arcs of increased brightness.

Húsavik (North Iceland I)

The next few weeks, I will write about this year’s vacation in Iceland’s north. For comparison, here are the links to the blog posts about Iceland’s south from two years ago:

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This year we stayed in Húsavik, a small and peaceful town a few degrees south of the polar circle.

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It lies on the east shore of Skjálfandi bay, which allows for nice sunsets (unless it is too cloudy (often) or not cloudy enough (rarely).

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Dramatic clouds are abundant and make driving dangerous.

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A highlight was the full moon backlit with a setting sun at midnight.

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Over and Under

Time for a game. You will need one or more decks of the following 16 triangular cards.

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The first game is a puzzle and asks to tile a triangle with all cards from a complete deck so that the tiles match along their edges, like so,

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except that in my attempt above two triangles don’t match. No hints today.

Next we use one or more decks to play a 2-person game. All cards are shuffled and form a single deck, top card visible. You will need three special cards, each just marked with one of the three card colors on it. Both players draw one of these color cards and keep their color secret. 

Now they take turns picking the top card from the deck and placing it on the table so that it matches previously placed cards. However, this time the matching has to happen along half-edges. For instance, after using a 16 card deck, the table might look like this:

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Each newly placed card has to border a previously placed card, and match along half edges on all sides where it borders another card.

When all cards are played, the players reveal their secret colors and score. Note that the colored arcs form chains of equal color. Suppose that player A is orange and player B purple. A looks at all orange chains of length at least 2. There are three orange chains of length 2 and one of length 3. For each of these chains, A counts how often they go over a purple arc. This happens 7 times, and A scores as many points. Similarly, B looks at all purple chains of length at least two. There are three, of lengths 2, 3, and 4. They go over an orange arc six times, so A wins by one point.

The idea is to arrange cards in chains of your color that go often over the snakes of your opponent’s color. The problem is, of course, that in the beginning you won’t know your opponent’s color. So you might want to put cards that have your arc go under both other arcs not into chains but out of the way. This, however, might give away your color…

Finally, here is a 3 person game played on a hexagonal board of edge length 4.

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The three players are dealt a color card each, and again the colors are being kept secret. You will need six decks of cards, for a total of 96 cards. Shuffle all cards and let each player grab 32. Then the players put their cards onto the board so that they meet at least one previously placed card along an entire edge, and match the colors of all cards they meet. After all cards are played, the board will look similar to the one below, except that there will be crossings.

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When the board is completely tiled, the players reveal their colors and score: For each completed circle of their color, the player counts how often that circle stays above the other two colors. This can happen (for each circle) between 0 and 12 times. That number is squared, and all numbers for all full circles for each color are added up, giving the score for that player.

There is a little catch to be aware of: There are two colors with 12 full circles each, and one color with 13 circles (purple in the example). Clearly the player with 13 circles will have an advantage when scoring. The first player will decide which color has 13 circles, and is therefore likely to claim the advantage. But then the two opponents might unite…

After Dark (Southern Illinois IV)

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The trails at the Giant City State Park close at dusk. The reason seems obvious: You could get lost, fall off a cliff, and die.

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The truth is more sinister. After sunset, the innocent looking gnomes go into hiding, and from the rocks the true owners of the place emerge.

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These two above still have humanoid features, but you start wondering whether there are other nameless horrors here. Death by falling off a cliff might have been a merciful alternative.

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I am sure Howard Phillips Lovecraft would have found inspiration here.

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The Twisted Color Wheel

Colors are curious. Physics tells us that, like sound, they are just waves. But while we can hear sound waves of wave lengths between 17mm and 17m, the wave lengths of visible light range between 390 and 700 nm. That’s sad. Wouldn’t it be cool if we could see colors resonate by being able to see both red (700nm) and ultraviolet (350 nm). Of course matters are more complicated because we perceive colors differently, using three specific color sensors. Imagine being only able to hear three different sound frequencies.

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A side effect of our biological limitation is that a color wheel makes sense to us, i.e. a continuous arrangement of the colors around a circle so that antipodal points represent complementary colors. This gave me the idea that one could color a Möbius strip continuously by hue so that points in the “front” and “back” are colored by complementary colors. Here is a 7-fold twisted rectangle as a ruled surface,

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and here a minimal surface version based on a torus knot:

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Finally, a Klein bottle, the immersion being obtained by rotating and revolving a figure 8 curve:

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Inside (Southern Illinois III)

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The Giant City State Park has its name because of the sandstone that has eroded into blocks (houses), separated by streets. After walking around last time, today we enter.

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The whole place is quite spooky, even during daylight, and it is easy to get lost.

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Some paths have exists, fortunately.

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The location of the next two images was most impressive. It looks rather artificial, but not made by humans.

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Who would design something like this where even the trees contort in desperation?

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(to be continued)