Simplex.
The series of simplex polytopes defined by a singularity (zero dimensions) and the repeated process of adding a vertex one unit distance from all the exisiting vertices in the positive direction of a dimension orthogonal to all the existing dimensions has vertex co-ordinate n-tuples {…, [2(n-1)(n-2)]-½, [2n(n-1)]-½, [(n+1)/2n]-½, 0, 0, …}.