Extracta Mathematicae (Jun 2019)
Tetrahedral chains and a curious semigroup
Abstract
In 1957 Steinhaus asked for a proof that a chain of identical regular tetrahedra joined face to face cannot be closed. Świerczkowski gave a proof in 1959. Several other proofs are known, based on showing that the four reflections in planes though the origin parallel to the faces of the tetrahedron generate a group R isomorphic to the free product Z2 ∗ Z2 ∗ Z2 ∗ Z2 . We relate the reflections to elements of a semigroup of 3 × 3 matrices over the finite field Z3 , whose structure provides a simple and transparent new proof that R is a free product. We deduce the non-existence of a closed tetrahedral chain, prove that R is dense in the orthogonal group O(3), and show that every R-orbit on the 2-sphere is equidistributed.
