Tetrahedral chains and a curious semigroup

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Tetrahedral chains and a curious semigroup

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Title: Tetrahedral chains and a curious semigroup
Author: Stewart, Ian
Abstract: In 1957 Steinhaus asked for a proof that a chain of identical regular tetrahedra joined face 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 the tetrahedron generate a group ℛ isomorphic to the free product ℤ₂ ∗ ℤ₂ ∗ ℤ₂ ∗ ℤ₂. We relate the reections to elements of a semigroup of 3 X 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.
URI: http://hdl.handle.net/10662/9982
Date: 2019


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