  In this paper we study a class of algebras having $n$-dimensional pyramid
shaped quiver with $n$-cubic cells, which we called $n$-cubic pyramid algebras.
This class of algebras includes the quadratic dual of the basic $n$-Auslander
absolutely $n$-complete algebras introduced by Iyama. We show that the
projective resolution of the simples of $n$-cubic pyramid algebras can be
characterized by $n$-cuboids, and prove that they are periodic. So these
algebras are almost Koszul and $(n-1)$-translation algebras. We also recover
Iyama's cone construction for $n$-Auslander absolutely $n$-complete algebras
using $n$-cubic pyramid algebras and the theory of $n$-translation algebras.
