The taxicab space group
type: Item
instance of: scholarly article
title: The taxicab space group
zbMATH Open document ID: 1199.51002
author: Özcan Gelişgen
author: Rüstem Kaya
published in: Acta Mathematica Hungarica
publication date: 2009-12-29T00:00:00Z
review text: The 3-dimensional taxicab space is defined as the space \(\mathbb R^3\) with distance function \(d(X,Y):=| x_1-y_1| +| x_2-y_2| +| x_3-y_3| \). In this geometry, the unit ball is the regular octahedron \(\mathcal O\) with (six) vertices \((1,0,0)\), \((0,1,0)\), \((0,0,1)\), \((-1,0,0)\), \((0,-1,0)\) and \((0,0,-1)\). It is almost obvious that the symmetries of the unit ball relative to \(d\)-distance correspond exactly with the symmetries of the regular octahedron in view of the Euclidean metric. So, the group of isometries of the 3-dimensional taxicab space is the semi-direct product of the octahedral group \(S(\mathcal O)\) (the orthogonal group) and the group \(T\) of all translations of \(\mathbb R^3\).
reviewed by: Erhard Quaisser
Mathematics Subject Classification ID: 51B20
Mathematics Subject Classification ID: 51F99
Mathematics Subject Classification ID: 51K05
Mathematics Subject Classification ID: 51N25
zbMATH DE Number: 5651936
zbMATH Keywords: isometry
zbMATH Keywords: taxicab metric
zbMATH Keywords: taxicab geometry
zbMATH Keywords: 3-dimensional
zbMATH Keywords: octahedral group
zbMATH Keywords: orthogonal group
MaRDI profile type: Publication
full work available at URL: s10474-008-8006-9
OpenAlex ID: W1997391628
cites work: Q4670650
cites work: Q5474644
cites work: Mappings preserving some geometrical figures
cites work: On mappings preserving pentagons
cites work: The Taxicab Group
DOI: 10.1007/S10474-008-8006-9
description: scientific article; zbMATH DE number 5651936