  The family of translation surfaces $(X_g,\omega_g)$ constructed by Arnoux and
Yoccoz from self-similar interval exchange maps encompasses one example from
each genus $g$ greater than or equal to $3$. We triangulate these surfaces and
deduce general properties they share. The surfaces $(X_g,\omega_g)$ converge to
a surface $(X_\infty,\omega_\infty)$ of infinite genus and finite area. We
study the exchange on infinitely many intervals that arises from the vertical
flow on $(X_\infty,\omega_\infty)$ and compute the affine group of
$(X_\infty,\omega_\infty)$, which has an index $2$ cyclic subgroup generated by
a hyperbolic element.
