4-regular vertex-transitive tilings of \(\mathbb{E}^3\) (Q1580750)

In an attempt to find tilings that carry the 18 uninodal zeolite structures, the authors introduced the concept of an extetrahedron of level \(h\), which is obtained by inserting new vertices of degree 2 into some of the original edges, up to \(h\) in each edge. Using the methods of \textit{O. Delgado Friedrichs} and \textit{D. H. Huson} in Discrete Comput. Geom. 21, No. 2, 299-315 (1999; Zbl 0927.52031), the authors prove the following theorem: There exist precisely 149 topological types of semipolytopal tile-transitive tilings of three-dimensional Euclidean space by extetrahedra of level 1, of which exactly nine are by combinatorial regular tetrahedra. Dualization gives rise to 149 types of 4-regular vertex-tilings. If tilings are not required to be semipolytopal, then exist 1571 further types.