Tetrahedral chains and a curious semigroup

• Ian Stewart ^[1]
  1. [1] University of Warwick

     University of Warwick

     Reino Unido

     
• Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 34, Nº 1, 2019, págs. 99-122
• Idioma: inglés
• DOI: 10.17398/2605-5686.34.1.99
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• Resumen

  • 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 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.