  Meier and Zupan proved that an orientable surface $\mathcal{K}$ in $S^4$
admits a tri-plane diagram with zero crossings if and only if $\mathcal{K}$ is
unknotted, so that the crossing number of $\mathcal{K}$ is zero. We determine
the minimal crossing numbers of nonorientable unknotted surfaces in $S^4$,
proving that $c(\mathcal{P}^{n,m}) = \max\{1,|n-m|\}$, where
$\mathcal{P}^{n,m}$ denotes the connected sum of $n$ unknotted projective
planes with normal Euler number $+2$ and $m$ unknotted projective planes with
normal Euler number $-2$. In addition, we convert Yoshikawa's table of knotted
surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number
for each surface in the table and providing upper bounds for the crossing
numbers.
