  We use first principles simulations to investigate the electronic properties
of a set of carbon nanocages with a bipartite structure. These nanocages are
exclusively formed by hexagonal and tetragonal rings and we show they feature
frontier states with particular symmetries as dictated by well defined rules
based on the size of the structure. We also show that spin-polarized
configuration emerge at the edges of open versions of these 0D systems. These
cages are further proposed as elementary building blocks for periodic 1D and 2D
systems. Even though we find different ways to define the particular bonds
making the connection between the cage-like units, these systems always show a
semiconducting behavior, both for 1D and 2D cases. However, the details of
linking hierarchies interfere in the degree of localization of the frontier
states of these crystalline systems.
