= Mid-edge facet centroid invariance
{title2=$\overline p^{\,(r+1)}=\overline p^{\,(r)}$}
The descendant of a face with cyclic vertices $p_0,\ldots,p_{n-1}$ has vertices $p_i'=(p_i+p_{i+1})/2$. 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.
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