  We give a rigorous definition of the T-fractal translation surface, and
describe some its basic geometric and dynamical properties. In particular, we
study the singularities attached to the surface by its metric completion and
show there exists a Cantor set of "elusive singularities." We show these
elusive singularities can be thought of as a generalization of the wild
singularities introduced by Bowman and Valdez. In particular, we show that
every elusive singularities has an infinite discrete set of rotational
components.
