Polynomial generation by regular Loop subdivision

For the three unit-grid edge vectors $v_1,v_2,v_3$ and a multivariate polynomial $f$ of total degree at most three, sampled regular-grid Loop subdivision generates the graph of $f+\frac{1}{12}{\sum }_{r=1}^3(v_r\cdot \nabla {)}^{2}f$. Thus affine polynomials are reproduced exactly and all polynomials through degree three retain their degree. Fourth-degree data constant along one edge direction reduce to a Cardinal cubic B-spline, whose limit need not be a polynomial.