We introduce a new characterization of linear isometries. More precisely, we prove that if a one-to-one mapping f:ℝn→ℝn(2≦n<∞) maps every regular pentagon of side length a> 0 onto a pentagon with side length b> 0, then there exists a linear isometry I :ℝn→ℝnup to translation such that f(x) = (b/a) I(x).
It is shown that the group of isometries of the 3-dimensional space with respect to taxicab metric is the semi-direct product
of octahedral group Oh and T(3), where Oh is the (Euclidean) symmetry group of the regular octahedron and T(3) is the group of all translations of the 3-dimensional space.
, or the
Bridge of Asses
, refers to Proposition 5 of Book I of Euclid’s
. This proposition and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal. Analogues of these propositions for higher dimensional
-simplices are considered in this paper, and satisfactory results are obtained for orthocentric
-simplices. These results do not hold for non-orthocentric
-simplices, thus supporting the point of view that orthocentric
-simplices and not arbitrary ones are the adequate generalization of triangles.