Planes of Leibniz, Lines of Weierstrass, Varia

Abstract

Take two distinct points p, q of ℝ³, i.e. p, q ε ℝ³, p ≠ q, and collect all s ε ℝ³ such that the euclidean distance of p, s and that of q, s coincide. The result will be a plane of ℝ³. This simple and great idea of Gottfried Wilhelm Leibniz (1646–1716) allows us to characterize hyperplanes of euclidean, of hyperbolic geometry, of spherical geometry, the geometries of Lorentz–Minkowski and de Sitter through the (finite or infinite) dimensions ≥ 2 of X as will be shown in the present chapter.