  We use a variation on Mason's $\alpha$-function as a pre-dimension function
to construct a not one-based $\omega$-stable plane $P$ (i.e. a simple rank $3$
matroid) which does not admit an algebraic representation (in the sense of
matroid theory) over any field. Furthermore, we characterize forking in
$Th(P)$, we prove that algebraic closure and intrinsic closure coincide in
$Th(P)$, and we show that $Th(P)$ fails weak elimination of imaginaries, and
has Morley rank $\omega$.
