  Some of the three-dimensional (3D) crystal structures are constructed by
stacking two-dimensional (2D) layers. It remains unclear whether this geometric
concept is related to the stability of ordered compounds and whether this can
be used to computational materials design. Here, using first principles
calculations, we investigate the dynamical stability of copper-based compounds
Cu$X$ (a metallic element $X$) in the B$_h$ and L1$_1$ structures constructed
from the buckled honeycomb (BHC) structure and in the B2 and L1$_0$ structures
constructed from the buckled square (BSQ) structure. We demonstrate that (i) if
Cu$X$ in the BHC structure is dynamically stable, those in the B$_h$ and L1$_1$
structures are also stable. Although the interrelationship of the metastability
between the BSQ and the 3D structures (B2 and L1$_0$) is not clear, we find
that (ii) if Cu$X$ in the B2 (L1$_0$) structure is dynamically stable, that in
the L1$_0$ (B2) is unstable, analogous to the metastability relationship
between the bcc and the fcc structures in elemental metals. The total energy
curves for Cu$X$ along the tetragonal and trigonal paths are also investigated.
