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The printed diagram is a propagation subdivision mask: it shows the influence of one old vertex on several new vertices. Its entries do not all belong to a single averaging stencil. The old vertex is retained, and a new vertex is inserted at each incident face centre. Reading the contributions arriving at a target givesThus each new vertex-vertex averages its old value and its four edge neighbors, while each new face-vertex is the centroid of its four old corners. Both stencils are convex combinations, reproduce constants and commute with affine maps. The neighboring $1/8$ contributions go to old vertices; the diagonal $2/8$ contributions go to face centres. The propagation coefficients sum to two because one refinement doubles the number of lattice vertices, not because either stencil fails to sum to one.

For calculations, describe this quincunx subdivision usingThe refined square lattice is rotated by $45^{\circ }$ and its spacing is reduced by $\sqrt{2}$. In its integer coordinates, the one-step mask isIndeed, a unit step in these new coordinates is an old face-centre displacement, whereas a diagonal step reaches an old edge neighbor. The even parity class contains the central and four diagonal coefficients and sums to one; the odd class contains the four axial coefficients and also sums to one. This checks the two target stencils independently.

Write the Laurent generating function of the one-step subdivision mask asFor quincunx subdivision, $p_j^{(1)}={\sum }_i{a}_{j-Di}{p}_i$. A second step gives $p_k^{(2)}={\sum }_{i,j}{a}_{k-Dj}{a}_{j-Di}{p}_i$. Setting $r=j-Di$ and $s=k-Dj$, its fine-grid offset is $k-2i=s+Dr$, since $D^2=2I$. Hence the binary mask is the convolution of $a$ with its $D$-transformed copy:Expanding gives the centered binary propagation subdivision mask, with row and column offsets running from $-3$ to $3$:The numerator entries sum to $256$, so $B(1,1)=4$, appropriate for a binary surface refinement. More specifically, each of the four parity classes sums to one after division by $64$, checking constant reproduction at every type of new lattice point.

A regular quincunx subdivision step replaces a square lattice by the union of its old vertices and face centres. In lattice indices one may useThus two steps give ordinary binary refinement. If $A(z,w)$ is the one-step subdivision mask symbol, the two-step symbol is $A(z,w)A(zw,z/w)$. The two lattice parity classes separately determine the old-vertex and face-vertex stencils; their coefficient sums must each be one to reproduce constants.

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.