Please use this identifier to cite or link to this item: http://hdl.handle.net/10662/9982
Title: Tetrahedral chains and a curious semigroup
Authors: Stewart, Ian
University of Warwick. United Kingdom
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.
URI: http://hdl.handle.net/10662/9982
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