  We study Lagrangian embeddings of a class of two-dimensional cell complexes
$L_{p,q}$ into the complex projective plane. These cell complexes, which we
call pinwheels, arise naturally in algebraic geometry as vanishing cycles for
quotient singularities of type $\frac{1}{p^2}(pq-1,1)$ (Wahl singularities). We
show that if a pinwheel admits a Lagrangian embedding into $\mathbf{CP}^2$ then
$p$ is a Markov number and we completely characterise $q$. We also show that a
collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\ldots,N$, cannot be
made disjoint unless $N\leq 3$ and the $p_i$ form part of a Markov triple.
These results are the symplectic analogue of a theorem of Hacking and
Prokhorov, which classifies complex surfaces with quotient singularities
admitting a $\mathbf{Q}$-Gorenstein smoothing whose general fibre is
$\mathbf{CP}^2$.
