Showing and Hiding (Spheres XI)

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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.

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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.

Reflections (Spheres I)

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Large scale mirrors like the surface of a lake are awe inspiring. They simultaneously create complexity and
order. The order comes from the inherent symmetry, and the complexity from subtle differences between original and
mirror image.

Things get considerably more complicated when the mirrors are curved. The Cloud Gate sculpture by Anish Kapoor (the Bean) in the Millennium Park in Chicago is a popular example. The multiple reflections create an immediately surprising chaotic richness of the reflection: Taking one step to the side changes the appearance of the reflection dramatically. But the sculpture also extends and therefore enriches the architecture.

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Motivated by this, I began to experiment with the spherical mirrors, spheres being the simplest curved shapes.
For multiple spheres touching each other there is a surprising phenomenon that is best understood when we begin with seven spheres of equal size, one at the center, and the remaining six surrounding the central sphere symmetrically. Complete this configuration by adding two planes that touch all seven spheres. Now pretend that the two planes are in fact also gigantic spheres. Than these two and the central sphere all touch the remaining six spheres, which in turn form a chain where consecutive spheres touch.

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It turns out that this picture is not just an approximation that only works in the ideal situation shown above where the big spheres are planes, but in fact works for spheres of any size. This is the content of Soddy’s theorem.

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To turn this into some sort of virtual sculpture, it is best to make just one of the spheres a plane. Then place two spheres onto the plane so that they touch. If you continue placing more spheres onto the plane so that they also touch the two initial spheres and the previously placed sphere, they will form a chain of six spheres of which the last again touches the first.

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Now imagine these being really large, reflective, slightly translucent, and illuminated with colored light sources. You might see something like this:

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This is the first of a series of images featuring ray traced spheres.