It is not possible to make a chain of regular tetrahedra that meet face to face and have the final one meeting up exactly with the first one. But one can come close.

This Demonstration lets you explore various chains of tetrahedra, all of which have edge-length 1. The integers defining the moves correspond to reflections in the four faces. Thus if neighboring integers are the same, they cancel. Without loss of generality we can always start a sequence with 12, and since the sequence 121212… does not lead to anything interesting, we may assume a sequence has 123 in it somewhere, and hence that it starts with 123. Thus the controls let you choose only for terms from the fourth to the eleventh.

There are several particular sequences that lead to loops that come close to closing up. The discrepancy from perfection is the maximum distance between a vertex of the first tetrahedron and a vertex of the best reflection of the last tetrahedron.

A Steinhaus chain is a sequence of regular tetrahedra such that consecutive ones meet face to face. In 1958 Świerczkowski proved that such a chain cannot end in a tetrahedron that is equal to the first one; the key point of the proof is that the group generated by the four face reflections is a free product of four copies of . But one can search for chains that approximate a perfect loop and this Demonstration shows several particular chains that have a small gap at the end. These were all found by the authors, except for the 96-loop, due to Robert Mathieson. The controls also allow for exploring various sequences of length 11. The tetrahelix, also called the Boerdijk–Coxeter helix or the Bernal spiral, is a noteworthy spiral of tetrahedra and arises from the sequence 12341234… . The fact that there are loops that come very close to closing up perfectly is evidence for the following conjecture.

Conjecture. For every there is a Steinhaus chain that has no self-intersections and is within ϵ of closing up perfectly.

References

[1] S. Wagon, Closing a Platonic Gap, The Mathematical Intelligencer, to appear.

[2] S. Wagon, The Banach–Tarski Paradox, New York: Cambridge University Press, 1985.