= Twice-smoothed four-direction box spline
Take two copies of each direction $(1,0),(0,1),(1,1),(1,-1)$. The resulting <box spline> has total polynomial degree six and $C^4$ continuity: there are eight directions and a rank-deficient remaining set has at most two, so the removal number is six. With a centred origin its support is the <zonotope>
$$
\{|x|\leq3,\ |y|\leq3,\ |x|+|y|\leq4\}.
$$
Its centred binary symbol is
$$
\frac{(1+z)^2(1+w)^2(1+zw)^2(1+z/w)^2}{64z^3w}.
$$
This symbol factors as $A(z,w)A(zw,z/w)$ with $A=(2+z+z^{-1})(2+w+w^{-1})/8$, giving an efficient <quincunx subdivision> implementation. The <box spline> criterion is also described in the primary research paper https://cims.nyu.edu/gcl/papers/velho20014s.pdf[4–8 Subdivision] by Velho and Zorin.
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