Is Delaunay triangulation unique for a given set of points

Question

For a given set of points in the 3D space or in a 2D space, can two different triangulation that conform to the Delanuay rules of empty circumcirle be created? If yes what would it depend on, the starting point?

Answer 1

A Delaunay triangulation is unique iff the circumcircle of every triangle contains exactly three points on its circumference: the vertices of the triangle.

For instance, the Delaunay diagram of the four vertices of a square is a square, and can be converted into a triangulation in two different ways.