  Let $\delta_0(P,k)$ denote the degree $k$ dilation of a point set $P$ in the
domain of plane geometric spanners. If $\Lambda$ is the infinite square
lattice, it is shown that $1+\sqrt{2} \leq \delta_0(\Lambda,3) \leq (3+2\sqrt2)
\, 5^{-1/2} = 2.6065\ldots$ and $\delta_0(\Lambda,4) = \sqrt{2}$. If $\Lambda$
is the infinite hexagonal lattice, it is shown that $\delta_0(\Lambda,3) =
1+\sqrt{3}$ and $\delta_0(\Lambda,4) = 2$. All our constructions are planar
lattice tilings constrained to degree $3$ or $4$.
