Month: September 2015

Avoiding Collisions (Helices I)

One of the simplest line configurations in space just utilizes the parallels to the coordinate axes that pass through the (red) points with integer coordinates.

Linepack

If we want to avoid the triple collisions at all these points, we can shift the lines one half unit each, like so:

Linepack2

This results in a dense packing of cylinders. Another possibility to avoid the collision is to let the lines spiral around the red points. I haven’t found a nice way to do this because the three helices would need to pass through the eight cubes surrounding a red point, meaning this is impossible in a symmetric way.

Linepack3

However, there is another line configuration where the lines pass through all the main diagonals. This is more complicated, because we have now four sets of parallel lines. Again we can shift the lines to avoid collisions.

Linepack6

Now, with four lines through each intersection, we can replace them by helices in a pretty symmetric fashion.

Spiralpack2

Displacement

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The meaning of the word Muscatatuck is not clear. According to Michael McCafferty’s book Native American Place Names of Indiana, it has its origins possibly in the Munsee words for swamp and river, or in the Lenape word for clear river. Both these languages were spoken by the Delaware, who migrated through this area in the early 19th century after continuous displacements by European settlers.

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These days, the name honors the Muscatatuck National Wildlife Refuge near Seymour. It is indeed a swampy place, and temporary home for many migratory birds.

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The pictures here are from a late afternoon visit while the weather was preparing for a storm. This didn’t leave much time for wildlife observations, but the barren landscape itself was well worth it.

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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.
Reddragon 01

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