Can we actually have null curves in Minkowski space?

Regarding null curves in flat space, how about $$ X(t) = (t,x,y) = (\tau, \cos(\tau), \sin(\tau)) . $$ Then $$ V(t) = (\dot t, \dot x, \dot y) = (1,-\sin(\tau), \cos(\tau)) $$ in which case $V^2 = 0$.

Any particle moving in $\mathbb{R}^3$ along any curve with constant speed $|v|=c$ will trace a null curve in Minkowski space.