= Quincunx subdivision
{title2=$D^2=2I$}
A regular <quincunx subdivision> step replaces a square lattice by the union of its old vertices and face centres. In lattice indices one may use
$$
D=\begin{pmatrix}1&1\\1&-1\end{pmatrix},\qquad D^2=2I.
$$
Thus 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.
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