摘要：In 1957 Steinhaus asked for a proof that a chain of identical regular tetrahedra joinedface to face cannot be closed. Swierczkowski gave a proof in 1959. Several other proofs are known,based on showing that the four reections in planes though the origin parallel to the faces of thetetrahedron generate a group R isomorphic to the free product Z2 Z2 Z2 Z2. We relate thereections to elements of a semigroup of 3 3 matrices over the nite eld Z3, whose structureprovides a simple and transparent new proof that R is a free product. We deduce the non-existenceof a closed tetrahedral chain, prove that R is dense in the orthogonal group O(3), and show thatevery R-orbit on the 2-sphere is equidistributed.
