The taxicab space group
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 \(\mathbb R^3\).
scientific article; zbMATH DE number 5651936