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.


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


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.


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.


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


Pebbles (Iceland X)

Djúpalónssandur has more to offer than spectacular rock formations.

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The desolate black beach consists of small, mostly smooth black pebbles

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some of which reveal, when submerged in water, a surprising colorization.

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This means many happy hours of collecting and sorting.

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When dried and stacked, everything returns to black and white.



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.


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.


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.


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.


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


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.


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.


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)

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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.

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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:


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.

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


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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