Abstract Insertion of a 2-connected vertex into alternate edges of a 2D net leads to a new family of 3D nets. The simplest operation is conversion of 2-connected vertices of adjacent parallel 2D nets into a square-planar vertex. Relative rotation of tetrahedral vertices above and below each square-planar vertex converts it into a tetrahedral vertex. The simpler nets of each type are enumerated for the 63, 3.122, 4.82 and 4.6.12 2D nets; these nets have higher symmetry and less geometrical distortion than the infinity of nets which are not listed. A further set of nets is obtained by applying a sigma transformation in the plane of each original 2D net. The structures of the beryl and milarite groups of minerals are based on 3D nets obtained from the 63 net, and that of steacyite from the 4.82 net.