Please use this identifier to cite or link to this item:
Title: Tetrahedral chains and a curious semigroup
Authors: Stewart, Ian
Keywords: Tetrahedral chain;Free product;Semigroup;Density;Equidistribution;Spherical harmonic;Cayley graph;Cadena tetraédrica;Producto libre;Semigrupo;Densidad;Equidistribución;Armónico esférico;Gráfico de Cayley
Issue Date: 2019
Publisher: Universidad de Extremadura
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.
DOI: 10.17398/2605-5686.34.1.99
Appears in Collections:Extracta Mathematicae Vol. 34, nº 1 (2019)

Files in This Item:
File Description SizeFormat 
2605-5686_34_1_99.pdf470,3 kBAdobe PDFView/Open

This item is licensed under a Creative Commons License Creative Commons