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Tetrahedral Loops
Tetrahedral Loops
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.