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.


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)

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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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Bad Weather? (North Iceland XI)

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When the impenetrable fog clears, things become instantly dramatic again.

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An Essential Element (North Iceland X)

Like Dimmuborgir, Námaskarð is very close to Lake Myvatn. At the ground you see cracks and holes. Signs of otherwise rare creepy crawlies?

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Ominous bubble are visible in the little puddles.

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When you lift your head, you quickly lose all hope for discovering new life forms.

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You nose will have told you long ago that this is not paradise:

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We are in the middle of highly active sulfur springs.

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This is a fun place to be, briefly.

Walking on or Staying? (North Iceland IX)

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Dimmuborgir (Dark Castles) is a lava field near the much more mellow Mývatn area.

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Allegedly, here it is where Satan landed after his fall (or did he leave deliberately?).

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What makes the place spooky are not so much the bizarre rock formations but rather the cracks one can see everywhere: Are they just signs of decay, or an indication of growth and an immanent emergence of something else?

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At least, the holes and tunnels seem to always promise a way out.

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But what about those who got stuck and now have to look back at us, frozen in rock for millennia?

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In Between (North Iceland VIII)

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Between Dettifoss and Selfoss, the lava field landscape is often filled with rain and spray water. This contributes to a micro landscape with very little vegetation.

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Wide angle shots close to the ground

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create an illusion of being aerial photographs of a much larger landscape,

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in which the ground seems to float between endless water and sky.

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Maybe these are the dreams of our planet about future landscapes, to be built after we are all gone.

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White Noise (North Iceland VII)

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The two bridges up above and below over the Jökulsá á Fjöllum (which we saw in my recent post about Hljóðaklettar) are about 50km apart.

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In between, there is no reasonable way to cross the Jökulsárgljúfur canyon. The entire area is a gigantic flood plain, with the flooding occurring every few thousand years, and caused by volcanic eruptions under the Vatnajökull, the glacier that gives the National Park its name.

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In between the bridges are three enormous waterfalls. Let’s begin with the Hafragilsfoss.

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In this region, water seems to come from everywhere.

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A little further south is the Dettifoss. Below are pictures from both sides of the bank.

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It is impossible to convey the physical experience of the falling water in pictures.

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This is a perfect place to be quiet.

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Still further south, the Selfoss, much more mellow, but still powerful.

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Next time we’ll have a close look at the micro landscape between these falls.

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Currently, TripAdvisor has two reviews of the waterfall at Bólugil. One of them states the waterfall can be seen from the road, the other states they couldn’t find it. What you see from the road is a little less than what you see above, and, while pretty, not exciting.

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To find this waterfall, drive west on 1 from Akureyri, towards Varmahlíð. There will be a sign to the farm Bóla to the right, with a four-leaf clover symbol on it, indicating some sort of attraction. Park at the gate or drive through to the bridge. On the way, you’ll see a fenced in area with a monument for the poet Bólu-Hjálmar. Why would be a monument here, in the middle of nowhere?

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This is all very frustrating. From the little that is available online, this Bólu-Hjálmar must have been an interesting character, living here in Bóla in the late 19th century, and writing sarcastic poems. Our library has a biography about him, in Icelandic, which I regretfully don’t speak. But is says a lot about the Icelandic people and their dedication to literature that they put monuments up like this.

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Nature seems to have put up its own humorous dedication. When you reach the canyon, you get a hint that their might be more than the little stream. Climbing up on a possibly imagined path, there appear more little falls

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and more

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and more:

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European Waterfalls claims there are a total of seven steps. The way back to the car leaves time to meditate about the intrusion of civilization into this remote landscape.

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Hafrafell (North Iceland V)

According to Google, there are several mountains named Hafrafell in Iceland, and the one this post is about isn’t even on Google Maps. It’s barely a mountain anyway.
When driving east on 85 from Húsavík to Ásbyrgi, one can see a handful of small mountains that are often hiding in the clouds. Below is a typical view of the Hafrafell, a little over 500 meters high.

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On a less cloudy day they all promise good views of the landscape. We inquired at the ranger station at Ásbyrgi about that one above, and the extremely friendly ranger told us to drive to the nearest farm, ask the owners for permission, leave the sheep alone, and find our way up.

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The farm is located on 865, which is a right turn off 85 when driving north, directly after the pool in Lundur. I doubt that many people ever go there, but the owner didn’t seem surprised. There is a path that leads to a small lake (mainly used by the sheep, very cooperative), and from then on one needs to find one’s own way.

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We followed the north-east face to its southern end and then went up. It’s pretty steep, and there was still lots of snow.

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The top is a large plateau with unlimited access to the all the clouds you can wish for. And the views were indeed worth the effort.

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To the east, a vast and empty landscape opens up. Very tempting. I need to learn riding a horse to get around there.

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Finally, one can see see the Ásbyrgi canyon from above, together with the mandatory threatening weather.

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Equally fascinating as the large scale landscape is the smaller scale vegetation. Of course you can find soft moss and dancing birches elsewhere, too, but here they are part of the deal.

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We spent about three hours on this mountain, 90 minutes to get up, 30 minutes on the top, and another hour to get down again. If you feel inspired and want to try it, check for snow and weather conditions before you go.