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 zonotopeIts centred binary symbol isThis 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 4–8 Subdivision by Velho and Zorin.