This is an example of an infinite polyhedron with very high genus.
It is not quite regular. Although it is composed of only a single type of regular face (all squares), it contains two types of vertices. In this way it's something like an infinite cousin of the finite Rhombic Dodecahedron. This relationship is more than just coincidental since it can be generated with the operation sometimes called "runcination" as applied to either the space packing of rhombic dodecahedra, or to the packing of tetrahedra and octahedra. The result of that operation is a space packing consisting of regular tetrahedra, octahedra, rhombic dodecahedra, rhombic prisms and triangular prisms. Removing all the triangles and rhombs from the resulting packing yields this infinite polyhedron.
This polyhedron divides space into two regions one connecting the volumes of the tetrahedra, octahedra and triangular prisms; and the other region connecting all the rhombic dodecahedra and rhombic prisms. As shown above, a repeat unit is composed of eight triangular prisms surrounding an octahedron. Four repeat units meet at a tetrahedron. The large spaces between these units are in the shape of Rhombic Dodecahedra which are connected to each other via Rhombic Prism shaped passages.
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