  The Rauzy gasket is the attractor of a parabolic, nonconformal iterated
function system on the projective plane which describes an exceptional
parameter set in various important topological and dynamical problems. Since
2009 there have been several attempts to calculate the Hausdorff dimension of
the Rauzy gasket, which is a challenging problem due to the combination of the
parabolicity and nonconformality in the geometry. In this paper we settle this
question by proving that the Hausdorff dimension of the Rauzy gasket equals the
(projective) affinity dimension. The key technical result underpinning this is
a partial generalisation of work of Hochman and Solomyak to the
$\mathrm{SL}_3(\mathbb{R})$ setting, where we establish the exact dimension of
stationary (Furstenberg) measures supported on the Rauzy gasket. The dimension
results for both stationary measures and attractors are established in broader
generality and extend recent work on projective iterated function systems to
higher dimensions.
