Showing and Hiding (Spheres XI)


A long time ago, we have looked at Soddy’s Hexlet, where a chain of six spheres is interlinked with a chain of three spheres.

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There are variations of this. For instance, you can have two interlinked chains of four spheres each.

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The alert visitor will have noticed that I am only displaying halves of spheres. This is because it is easier to add the other halves on one’s mind instead of thinking them away in order to see what’s behind.

There is more. If you take a suitable chain of five spheres, you can fit 10 around and through, but you will need to make three turns until the chain closes. This means that the spheres will touch their immediate successors, but intersect the ones after one and two turns, respectively.


There still is more, of course, which we leave to the reader to explore. Finding these chains is not difficult, provided you do this in the 3-dimensional sphere, and place the spheres inside complementary tori with suitable radii.


Decay (Museum Hombroich II)

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There are more things to see and do at the Museum Island Hombroich than to visit the pavilions. Artists in residence produce landscape art, and concerts are given.

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I wonder how this sculpture has withered since I took these pictures in 1992. This one is part of a full circle of such outcroppings.

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Mechanical structures clearly without purpose alternate with objects that are equally clearly of daily importance but could as well be just pieces of art.

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An outdoor museum where the objects are exposed to the elements defies the usual purpose of a museum: the preservation of its artifacts.
Here at Hombroich the time has just been slowed down a bit, making it the main object to contemplate.

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Karana Mudra (Museum Hombroich I)

When the Cold War ended, a missile base near Neuss, Germany, became obsolete. The area was bought by the industrial real estate agent Karl Heinrich Müller, and turned into the Museum Island Hombroich.

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Visitors are greeted appropriately by an Asian statue, holding his hand in the Karana Mudra gesture to ban evil spirits.

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Meticulously landscaped by Bernhard Korte, the area is populated with small buildings (landscape chapels),
by Erwin Heerich that contain Asian or contemporary art, or just empty space.

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Soft glass roofs and narrow doors create a balance between diffuse and directed light.

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The geometric harshness of the buildings disappears in the fading light.

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Little Things

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When hiking the rugged trail at the bottom of Clifty Canyon, you are in the shade most of the day.
This might make you miss the little things on the way, like this offering of leaves on a ledge of the canyon wall.

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Often, wood and rock combine to natural still lives.

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Or rocks make shelter for the little people,

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offer drawing tablets for future artists,

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and resting grounds for the elderly.

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In Chains (Annuli I)

A cube can be sliced half so that the cross section is a regular hexagon, and this even in four different ways.


In particular, we can place regular hexagons into space so that the corners all have integer coordinates, and the hexagons face four different directions. This suggests to interlink the hexagons, for instance like so:


The mathematician immediately will ask to put as many of such linked hexagons into space, and this automatically drives the discovery of new structures or leads to connections with the already known.

In this case, it turns out that feeding one hexagon through the center of another is not so smart.


When the center spot is taken, any further hexagon through either of the first two must be placed by breaking the symmetry. Above we have threaded three hexagons through a horizontal red hexagons in a rotationally symmetric way, and placed a mirror image of this tetrad below. While these two pieces are not yet interlinked, they can together be translated as to create a very tight interlinked system of annuli (replacing the hexagons with smooth annuli).


This works in all directions, and being stuck somewhere within this tangle will look like this:




After the invasion of Granada in 1492 by King Ferdinand II and Queen Isabella I, things changed in Granada.


One can debate how liberal the emirate had been, but it certainly didn’t get better. Whenever choices become limited,life does not necessarily become easier.


Today, we have a choice. We can face reality with clarity, like the cathedral of Granada, built over the ruins of its former mosque.


Or we can prefer the blurred view over the remaining walls of the Alhambra.


For the moment, it’s maybe best to keep gardening.

Big Things

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Clifty Falls State Park in eastern Indiana, at the border to Ohio, features a large canyon with several waterfalls, about 20 meters tall.


This allows for views that are not very common elsewhere in Indiana. Usually, when your view is not blocked by trees, it is an endless plane and an endless sky, cut in half by a perfectly straight horizon. At Clifty Falls, you can look across, giving you back a sense of size.

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Or, if you dare, you can look down.

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It is somewhat ironic that the only reason lonely trees have a chance to grow this tall is the protection of the canyon they are trying to outgrow.

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