Genus: 3
Symmetry: Multi-directional
This model is perhaps the simplist of the infinite regular polyhedra. It's one of the three that can be considered platonic as all faces and edges can be transformed into each other instead of just its vertices.
This model is very interesting because it is not rigid even when tiled to infinity. Its movement has three degrees of freedom, one for each of the three coordinate axis. It's difficult to physically construct this model without it collapsing unless these DOF are locked down first by adding any three non-parallel square "braces" into parts of the lattice where they don't belong. A single such "impurity" in the crystal lattice is enough to entirely lock its flexibility in one direction.
Main Infinite Regular Polyhedra page.
Superliminal home page.