  The commonly used spatial entropy $h_{r}(\mathcal{U})$ of the
multi-dimensional shift space $\mathcal{U}$ is the limit of growth rate of
admissible local patterns on finite rectangular sublattices which expands to
whole space $\mathbb{Z}^{d}$, $d\geq 2$. This work studies spatial entropy
$h_{\Omega}(\mathcal{U})$ of shift space $\mathcal{U}$ on general expanding
system $\Omega=\{\Omega(n)\}_{n=1}^{\infty}$ where $\Omega(n)$ is increasing
finite sublattices and expands to $\mathbb{Z}^{d}$. $\Omega$ is called
genuinely $d$-dimensional if $\Omega(n)$ contains no lower-dimensional part
whose size is comparable to that of its $d$-dimensional part. We show that
$h_{r}(\mathcal{U})$ is the supremum of $h_{\Omega}(\mathcal{U})$ for all
genuinely two-dimensional $\Omega$. Furthermore, when $\Omega$ is genuinely
$d$-dimensional and satisfies certain conditions, then
$h_{\Omega}(\mathcal{U})=h_{r}(\mathcal{U})$. On the contrary, when $\Omega(n)$
contains a lower-dimensional part, then
$h_{r}(\mathcal{U})<h_{\Omega}(\mathcal{U})$ for some $\mathcal{U}$. Therefore,
$h_{r}(\mathcal{U})$ is appropriate to be the $d$-dimensional spatial entropy.
