  We propose a class of models which generate three-dimensional random volumes,
where each configuration consists of triangles glued together along multiple
hinges. The models have matrices as the dynamical variables and are
characterized by semisimple associative algebras A. Although most of the
diagrams represent configurations which are not manifolds, we show that the set
of possible diagrams can be drastically reduced such that only (and all of the)
three-dimensional manifolds with tetrahedral decompositions appear, by
introducing a color structure and taking an appropriate large N limit. We
examine the analytic properties when A is a matrix ring or a group ring, and
show that the models with matrix ring have a novel strong-weak duality which
interchanges the roles of triangles and hinges. We also give a brief comment on
the relationship of our models with the colored tensor models.
