We start with a generic n-gon $$Q_0$$ Q 0 with vertices $$q_{j,0}$$ q j , 0 ( $$j = 0, \dots , n-1$$ j = 0 , ⋯ , n - 1 ) in the d-dimensional Euclidean space $${\mathbb {E}}^d$$ E d . Additionally, $$m+1$$ m + 1 real numbers $$u_0, \ldots , u_m \in {\mathbb {R}} \, (m < n)$$ u 0 , … , u m ∈ R ( m < n ) with $$\sum _{\mu = 0}^m u_\mu = 1$$ ∑ μ = 0 m u μ = 1 are given. From these initial data we iteratively define generations of n-gons $$Q_k$$ Q k in $${\mathbb {E}}^d$$ E d for $$k \in {\mathbb {N}}$$ k ∈ N with vertices $$q_{j,k}:= \sum _{\mu = 0}^m u_\mu \, q_{j+\mu , k-1}$$ q j , k : = ∑ μ = 0 m u μ q j + μ , k - 1 . We can show that this affine iteration generally regularizes in an affine sense.