Mid-edge facet centroid invariance (${\overline{p}}^{(r+1)}={\overline{p}}^{(r)}$)

The descendant of a face with cyclic vertices $p_0,\dots ,p_{n-1}$ has vertices $p_i^{\prime }=(p_i+p_{i+1})/2$. Their arithmetic mean is unchanged. On the cyclic Fourier basis, a mode of frequency $j$ is multiplied by $(1+e^{2\pi ij/n})/2$. Every nonconstant mode has modulus less than one, so the descendant face shrinks to the original vertex centroid. This puts that centroid on the subdivision limit, without requiring the original face to be planar.