The infinite regular polyhedra are periodic structures which have all the properties of the platonic solids but being finite. Their faces are thus identical regular polygons and their vertices are symmetrical and identical (the same number of faces meet at each vertex).
Such a structure may be built using an elementary module whose number of tunnels defines the genius of the polyhedron.
Here are three examples discovered in 1926 by John Flinders Petrie and H.S.M. Coxeter.
the most simplest is not rigid


lattice of truncated octahedra with their square faces removed


lattice of truncated tetrahedra with their triangular faces removed


In the sixties J.R. Gott and A.F. Wells published lists of pseudopolyhedra whose vertices are only identical (without being symmetrical). Here is my favorite:  
lattice of icosahedra linked with octahedral tubes


references:  Infinite regular polyhedra: Vladimir Bulatov  superliminal.com 
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