  We develop tools to study the topology and geometry of self-affine fractals
in dimension three and higher. We use the self-affine structure and obtain
rather detailed information about the connectedness of interior and boundary
sets, and on the dimensions and intersections of boundary sets. As an
application, we describe in algebraic terms the polyhedral structure of the six
fractal three-dimensional twindragons. Only two of them can be homeomorphic to
a ball but even these have faces which are not homeomorphic to a disk.
