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) |
Archivos
Archivo | Descripción | Tamaño | Formato | |
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2605-5686_34_1_99.pdf | 470,3 kB | Adobe PDF | Descargar |
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