I have been reading David Speyer's Perfect Matchings and the Octahedron Recurrence, trying to carry out his "cross-wrenches" generalization of the Aztec diamond. In what follows, I'm asking for a construction of $G_{n_0,i_0,j_0}$ in the case where the infinite graph $\mathcal{G}$ gives square-octagon tiling.

I found Speyer's notation very difficult. Maybe section 4 of Perfect Matchings and Perfect Powers by Mihai Ciucu will be easier to use for this special case.

In section 1.2 the "Aztec Diamond" theorem is stated $f(n_0, i_0, j_0) = \sum m(M)$

- $f(n_0,i_0,j_0)$ is the solution to the octahedron recurrence. $$ f(n,i,j)f(n-2,i,j)= f(n-1,i-1,j)f(n-1,i+1,j)-f(n-1,i,j-1)f(n-1,i,j+1)$$
- The sum over matchings of generalized Aztec diamond graphs $G(n_0, i_0, j_0)$ is called $\sum m(M)$. These graphs are embeeded in an infinite grid $\mathcal{G}$.
- For now, every matching is counted with unit weight: $m(M)=1$.
- $n_0+i_0+j_0 \equiv 0 \mod 2$

What is the sequence of shapes corresponding to Speyer's "crosses+wrenches" construction for the square-octagon lattice? Section 3.7 some relevant info.

The faces of his lattices are indexed by pairs of integers $(i,j) \in \mathbb{Z}^2$. He defines a "level" ( I really want to avoid using his word "height", which has another meaning in terms of dominos.)
$$ h(i,j) = \left\{ \begin{array}{rc} 0 & \text{if }(i,j) \equiv (0,0) \mod 2\mathbb{Z}\times 2 \mathbb{Z} \\\\
0 & \text{if }(i,j) \equiv (1,1) \mod 2\mathbb{Z}\times 2 \mathbb{Z} \\\\
1 & \text{if }(i,j) \equiv (0,1) \mod 2\mathbb{Z}\times 2 \mathbb{Z} \\\\
-1 & \text{if }(i,j) \equiv (1,0) \mod 2\mathbb{Z}\times 2 \mathbb{Z}
\end{array} \right.$$
The Octahedron recurrence has initial conditions $f(h(i,j),i,j)=1$ and specializes here to powers of **5**:
\begin{array}{rlc}
f(2n,i,j) & = 5^{n^2} & \\\\
f(2n+1,i,j) & = 5^{n^2+n} & \text{if } i \equiv n \mod 2\\\\
f(2n+1,i,j) & = 2 \cdot 5^{n^2+n} & \text{if } j \equiv n \mod 2
\end{array}
I'm not even sure this list of possibilities is exhaustive. For $(2n+1,i,j)$ only one of $i,j$ can be odd.

The shapes $G(n_0, i_0, j_0)\subset \mathcal{G}$ are planar graphs with "open" and "closed" faces. He defines the "lattice", "edges", and "faces":
\begin{eqnarray}
\mathcal{L} &=& \{ (n,i,j) \in \mathbb{Z}^2 : n = i + j \mod 2\} \\\\
\mathcal{E} &=& \{ (i,j) \in \mathbb{Z}^2: i + j \equiv 1 \mod 2\} \times \{ a,b,c,d\} \\\\
\mathcal{F} &=& \mathbb{Z}^2
\end{eqnarray}
The faces are indexed by pairs of integers. The edges are labelled **a,b,c,d**.

In section 2.1 Speyer defined some cones in $\mathbb{Z}^3$: \begin{eqnarray} p_{(n_0,i_0,j_0)} &=& n_0 - |i - i_0| - |j - j_0| \\\\ C_{(n_0,i_0,j_0)} &=& \{ (n,i,j) \in \mathcal{L}: n \leq n_0 - |i - i_0| - |j - j_0| \} \\\\ \mathring{C}_{(n_0,i_0,j_0)} &=& \{ (n,i,j) \in \mathcal{L}: n < n_0 - |i - i_0| - |j - j_0| \} \\\\ \partial C_{(n_0,i_0,j_0)} &=& \{ (n,i,j) \in \mathcal{L}: n = n_0 - |i - i_0| - |j - j_0| \} \\\\ \mathcal{I} &=& \{ (i,j, n) \in \mathcal{L}: n = h(i,j) \} \\\\ \mathcal{U} &=& \{ (i,j, n) \in \mathcal{L}: n > h(i,j) \} \end{eqnarray}

I have been unable to sort out definition of $G_{n_0,i_0,j_0}$ in section 3.3

It would be really amazing if one could show how the dominos actually "shuffle".