= 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 $f+\frac1{12}\sum_{r=1}^3(v_r\cdot\nabla)^2f$. 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.
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