We extend the first-passage time $T$ to $\x,\y\in\bR^2$ by setting $T(\x,\y):=T\big(\v(\x),\v(\y)\big)$, where $\v(\x)$ is the almost sure unique vertex $\v\in\calP$ with $\x\in\C_{\v}$. We say that $\rho(\v,\v')\in\calC(\v,\v')$ is a geodesic between $\v$ and $\v'$ if $t\big(\rho(\v,\v')\big)=T(\v,\v')$. For each $\x,\y\in\bR^2$ we denote $\rho(\x,\y):=\rho\big(\v(\x),\v(\y)\big)$. One can see that if $\bF$ is a continuous function then, almost surely, for all $\v,\v'\in\calD_v$ there exists a unique geodesic $\rho(\v,\v')$ [@p-105]. A self-avoiding and semi-infinite path $\rho=(\v_1,\v_2,\dots)$ in $\calD$ is called a semi-infinite geodesic if for all $\v_j,\v_k\in\rho$, the path $(\v_j,\v_{j+1},...,\v_k)$ is the unique geodesic connecting $\v_j$ to $\v_k$.

Given $k$ different points $\x_1,...,\x_k\in\bR^2$, the initial configuration of seeds, we define the multi-type growth process $\big\{(\B_{\x_1}(t),...,\B_{\x_k}(t))\,:\,t\geq 0\big\}$ by $$\B_{\x_j}(t):=\big\{\x\in\bR^2\,:\,\x\in c(\C_\v)\,\mbox{ for some }\,\v\in\calB_{\x_j}(t)\big\}\,,$$ where $$\calB_{\x_j}(t):=\big\{\v\in\calD_v\,:\,T(\x_j,\v)\leq t\,\mbox{ and }\,\min_{l=1,...,k}\{T(\x_l,\v)\}=T(\x_j,\v)\big\}\,,$$ and $c(\C_\v)$ denotes the closure of the tile $\C_\v$. If there exists $j<l$ such that $\v_{\x_j}=\v_{\x_l}$ then we set $\B_{\x_j}$ as before and $\B_{\x_l}(t)=\emptyset$.

When $k=1$ then we have a single growth process $\B_\x(t)$ which represents the set of points reached by time $t$ from the initial seed $\x$. For a continuous distribution $\bF$ satisfying the following shape theorem [@p-105; @VW92] holds: there exists a constant $\mu(\bF)\in(0,\infty)$, namely the time constant, such that for all $\epsilon>0$ $$\bP\big((1-\epsilon )t \D(1/\mu)\subseteq 