## Comparing two geometric problems ### Hexlet

Having solved Steiner's porism which involves rings of circles in the plane, it is natural to think if there are similar problems involving rings of spheres in the three dimensional space. It is from this context that Hexlet emerges. Though it is not a simple equivalent of the previous problem in the plane both problems share some similarities. We fix three pairwise tangent spheres with three distinct tangency points. If we add to those three spheres another sphere that is tangent to each of them, there will be in general7, "on each side" of this new surface, a unique sphere that is tangent not only to the previous one but also the three spheres fixed at the beginning. The choice of the first will, in general, determine a ring of spheres in which every sphere is tangent to the previous one and all spheres in the ring are tangent to the three spheres fixed initially. We say this ring is either closed or open depending on whether the last sphere is tangent to the first one or not, respectively.

In figures 16, 17 and 18 we see represented, respectively, a triplet of spheres – the red one containing the other two, blue and green –, the triplet together with a (closed) ring of six spheres and the ring, only.

In figures 19 and 20 we see represented, respectively, a triplet of pairwise exterior spheres and the triplet together with a yellow sphere8 containing all the others.

The analogous problems analysed previously in the plane, can be formulated in the following way: 1. Are there rings that close and rings that do not? 2. Does "Whether it closes or not" depend on the initial positions and sizes of the spheres but not on the position of the first sphere in the ring? 3. Does the number of spheres in a ring that closes varie depending on the radii and the position of the three initial spheres but not on the position of the first sphere in the ring? 4. Is "Not to close" the generic situation and "to close" the exceptional one?

7 In fact, there are limiting cases where the sphere degenerates into a plane.
8 Note that in figures 16, 17, 20 and 22 there is a semi-transparent coloured sphere containing the others which changed slightly their colours.

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