Nets with channels of unlimited diameter

Abstract

Three-dimensional nets and polyhedra1,2 present an unlimited mathematical challenge, and systematic enumeration of the infinite series of four-connected nets is just developing3–10 using selected algorithms. Several framework structures were solved and classified by invention of new nets (see refs 11–13). The four-connected nets of low density are of practical interest because their nodes may correspond with the tetrahedrally-coordinated atoms of aluminosilicate zeolites14, some silica polymorphs15, aluminophosphates16,17 and other molecular sieves. The nets can be classified by circuit symbols1–7 and coordination sequences18,19. Each oxygen atom of a zeolite and other tetrahedral frameworks lies near the midpoint of a branch connecting two tetrahedral nodes, and the empty volume between the close-packed oxygen atoms is important for molecular sorption. The largest ring in a natural zeolite or synthetic molecular sieve involves 12 tetrahedra11,14, but larger rings and sphere packings of lower density are known12,20–24. We report here two new infinite series of four-connected three-dimensional nets which contain channels of unlimited diameter.