  We show that there is a pair of smooth complex quartic K3 surfaces $S_1$ and
$S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are isomorphic as abstract
varieties but not Cremona isomorphic. We also show, in a geometrically explicit
way, that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$
in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are Cremona isomorphic, but not
projectively isomorphic. This work is much motivated by several e-mails from
Professors Tuyen Truong and J\'anos Koll\'ar.
