Hopf Fibration (Annuli III)

Hopf 8 right

When talking about tori, at some point the Hopf fibration will make its appearance.
It all begins with a few tori of revolution packed together. Think about circular wires
bundled into one thick cable.

Simple

Cut through all the wires, twist the cable by 360 degrees, and reconnect wires of equal color.

Twist

Now all wires are interlinked, and this has the advantage that you can extend all this wiring to all of space (except for the vertical axis) in an even way to het what mathematicians call a fibre bundle.

Side

One can increase the complexity by showing nested wires by removing parts of then. The top view below is a simplified version of the picture at the top.

Top

Djúpalónssandur (Iceland IX)

Djúpalónssandur is a rocky beach in the southwestern corner of the Snæfellsjökull National Park.

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Besides its historical significance of an old fishing port (of which only the remains of a few huts are visible), it features bizarre lava rock formations.

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The grassy slopes of the Snæfellsjökull seem to just break off into the sea, as if the landscape builder left his work unfinished.

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If fire could solidify, it would look like this.

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The 120-Cell (Spheres XIII)

Dodeca right

Pentagons do not tile the plane. If you fit three of them around a corner, there will be a gap of 36 degrees.
But, on a sphere, the pentagons can be inflated so that their angles become 120 degrees, and then twelve of them can be used to tile the sphere, creating a spherical version of the dodecahedron.

Dodeca spherical

Likewise, dodecahedra do not tile space. When you fit three around an edge, they leave a gap of about 10.3 degrees.
But again, they can be inflated in the 3-dimensional sphere. This time you will need 120 of them to tile the entire sphere. To visualize this, we start with one dodecahedron, and attach copies at opposite faces. After 10 copies, you will obtain an annulus of dodecahedra, which looks like this, after stereographic projection:

Dodeca

Repeat this with all immediately neighboring dodecahedra to get five more intertwined annuli of dodecahedra. They hide the original annulus from view. All six annuli together form one half of the 120 cell, the rest just being the complement in the 3-sphere of what we already have.

Dodeca 2

Here is an image of just the vertices and edges of the 120-cell. No elephants were harmed in making the 1200 ivory edges.

120cell

Hell (Iceland VIII)

After Plato had the brilliant idea to use a hypothetical reward system in an equally hypothetical afterlife as the ethical foundation of a functioning society, it didn’t take long until picturesque ideas about how the rewards might look like started to spread.

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Unsurprisingly, the focus was not so much on positive rewards like eternal bliss, but rather on the peculiarities of punishments.

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The Seltún Geothermal Area near Reykjavik provides at least the mandatory ambience of heat and stench.

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There are even indications of horned minions ready to pull you under. Clearly, the ground is treacherous here.

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Why is it that we take delight in all this unpleasantness?

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Reflections on Simplicity

If a parabolic mirror has a light bulb in its focal point, the light rays are reflected at the mirror into parallel light beams, evenly illuminating whatever lies ahead.

Parabola

In optics, reflections are well studied. The basic question is what happens when parallel light hits a reflective surface.

The case of the parabola is the rare exception. Typically, the reflected light rays will produce another curve of heightened brightness, called its caustic.
For instance, you might have observed a strangely formed curve in a cup of good black tea when horizontal light hits the rim of your circular cup. This curve is actually a nephroid, well studied since antiquity.

Circlecaustic

Doing the same to an ellipse gives a deformed picture.

Ellipsecaustic

Still other curves like the spiral below have elegantly ornamental curves as their caustics.

Spiralcaustic

Hraunfossar (Iceland VII)

Iceland has a lot of water falls. It is so bad that you shrug off the ones that would be worth a day trip at home, (almost) no matter where you live.

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Of the few that we saw this summer (in 2015), my favorite was not among the big ones.
We had just pulled into a parking lot by chance, and 3 minutes away from the road, I couldn’t but smile.

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This was not a single waterfall, but literally hundreds of little ones on the far side of the Hvítá river.
The falls originate from many separate springs in the lava field in the back.
It felt like the elves had been practicing here before they started to work on the big ones.

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Each single fall is a masterpiece that dances among companions.

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Hic Sunt Dracones

Take a long strip of paper, and fold it left over right, then left over right again, and so on, a couple of times.
Even if your strip is very thin and long, you probably won’t be able to do that more than six or seven times.

Then carefully unfold the paper so that each bent makes a right angle. What you get will look like this:

Samples 01

Another method leads to the same curves. Start with a curve consisting of two segments, making a right angle. Think of it as being a track you want to walk along. Things being difficult, you happen to swerve slightly to the right on the first segment, and on the second slightly to the left, meaning that instead of following the blue path, you walk the red path:

Drunk1 01

Now try again, this time starting with the red path that is four segments long (and colored blue below). The same happens, you alternatingly swerve right and left, creating the next (red) path. The curves will be the same ones as above.

Drunk 01

Is there any sense to it? Things get more amusing if you replace each segment by a square that has that segment as a diagonal. This turns the curves into polyominoes, as you can see for the first few cases below.

Polyominization 01

You will also see that these shapes start resembling a common dragon. If you keep folding a little while, more details emerge.
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But it gets better. All these polyomino-dragons tile the plane, interlocking perfectly. Both the young dragonlings

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and also the older, wrinkly ones:

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Now imagine stacking these dragons on top of each other, generation by generation. If I had the money, my mansion would look like that.

Dragonhouse

Leaves

In preparation of the often wondrous fall season in Indiana, here is a collection of leaves. Some pictures were taken over 20 years ago, like this one, taken at an early spring hike in the Kottenforst near Bonn.

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So we are back to the Event Horizon.

Here is another image of a leaf floating on water. While the bottom of the stream is a clear indication of looming decay, the blue reflection of the sky tells stories of greener days. Do trees need a religion?

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Poets often liken leaves to hands, and this in mind, the next image becomes quite creepy.

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So let’s return to the peaceful state of the beginning, when there is no time anymore.

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Black, Green, and White (Iceland VI)

Googling for Black, Green, and White leads to some interesting things. There is black, green, and white tea, of course (to my delight). There are black, green, and white wires that puzzle some hobby electricians. Then there are some countries that have these colors in their flags. Iceland should be one of them, but isn’t.

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Green is not only the color of the moss that covers the older lava fields. You have it with the algae

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and the shrubs,

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always contrasted by water, sky, and earth. The simplicity of the color pattern is contrasted by an astounding complexity and diversity of the landscape.

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Much of earth must have looked like that before man, and maybe will look like that again.

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Ramsey Cascades

In June 2009, I spent a week hiking in the Smoky mountains. There was much to see, and I will focus here on the Ramsey Cascades waterfall, in the northeastern part of the park.

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The four mile, 2,000′ climb with camera and tripod is strenuous, but as soon as you arrive at the fall, all the pain is forgotten. This is one magnificent waterfall.

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Waterfalls are tricky. Being there is obviously exciting, but not being there and instead having to look at pictures is annoying. So I apologize.

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Photographers have tried to get the most out of waterfalls. The rules of the game have become: Avoid direct sunlight, and use long time exposure to get the surrealistic filaments of water. This is supposed to turn any waterfall into a world of wonder.

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Don’t misunderstand me; I like alienation. But there are other ways, too. A waterfall has a personality that wants to be discovered and appreciated. The Ramsey Cascades are a wonderful example with a highly complex personality.

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So this post is a teaser, a puzzle. Instead of showing the entire fall, I only show closeups, highlighting the many different ways how water and rock interact.