Identificador persistente para citar o vincular este elemento: http://hdl.handle.net/10662/9982
Títulos: Tetrahedral chains and a curious semigroup
Autores/as: Stewart, Ian
Palabras clave: Tetrahedral chain;Free product;Semigroup;Density;Equidistribution;Spherical harmonic;Cayley graph;Cadena tetraédrica;Producto libre;Semigrupo;Densidad;Equidistribución;Armónico esférico;Gráfico de Cayley
Fecha de publicación: 2019
Editor/a: Universidad de Extremadura
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 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
DOI: 10.17398/2605-5686.34.1.99
Colección:Extracta Mathematicae Vol. 34, nº 1 (2019)

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