The Infinite Case
The "Trapped in Thickland" puzzle in Peter Winkler's latest edition of Mathematical Puzzles asks the following (in my own words):
The Infinite Asteroid Belt is a region in $\mathbb{R}^3$ between two parallel planes. It contains a (possibly infinite) number of pairwise-disjoint congruent convex bodies (i.e. convex compact sets with non-empty interior), called asteroids. Could it be the case that none of the asteroids can be taken out of the Infinite Asteroid Belt without colliding with any other asteroids, and without moving any other asteroid?
Let's call a set of asteroids satisfying this a "stable cluster". As spoiled by the title of this question, there exists an asteroid shape and an arrangement of infinitely many asteroids that forms a stable cluster. Here's some vertical space if you want to think about what that would look like.
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So here's the stable cluster given in Winkler's book. We take the asteroid shape to be a regular tetrahedron, and then arrange them to form the following pattern:
The construction has some flexibility. For instance, we can enlarge the tetrahedra to form a much tighter pattern.
The Finite Case
A natural next step is to ask whether finite stable clusters exist. That is, does there exist a bounded set $E \subseteq \mathbb{R}^3$ containing (finitely many) pairwise-disjoint congruent convex bodies, none of which can be moved out of $E$ without moving any other convex body and without collision?
Miraculously, I found that the answer is still "yes"! The previously displayed arrangement of tetrahedral asteroids has some room for flexibility, and in particular this means that the "plane" formed by the tetrahedra can be given some curvature to form a compact manifold.
However, it is unclear if the asteroids can be wrapped into a sphere (a natural first candidate) while maintaining their delicate interlocking structure. Which is why I instead wrapped them into a torus!
The monstrosity you see here is a stable asteroid cluster formed by $8000$ tetrahedra. For visual proof that this indeed works, here are photos of the "outer curvature" and the "inner curvature", respectively:
Neat! I can provide details on this construction upon request, though it should be fairly believable that such a construction cannot fail provided that the torus is large enough. Indeed, if the original "infinite" construction works, then surely it should still work after a small perturbation. So the toroidal arrangement should also work provided that, locally, the curvature is negligible enough to induce only small perturbations. Thanks to the choice of using a torus, the interlocking structure can be preserved for sure.
The Question
Last time I checked, $8000$ is a finite number. That means that there must exist a minimum required number of asteroids in $\mathbb{R}^3$ that can be arranged to form a stable asteroid cluster. What is this minimum?
Judging by the problem's nature, I find it quite unlikely that I will see a definitive answer to this within my lifetime, or within any of my next several reincarnations. Hence, I welcome any answer that can provide either upper or lower bounds on this minimum number. I'm quite interested in how much we can narrow this down.
