  We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov
model is dependent on four spin variables which are the linear combinations of
the spins on the corner sites of the cube and the Wu-Kadanoff duality between
the cube and vertex type tetrahedron equations is obtained explicitly for the
Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by
considering the symmetry property of the weight function, which is
corresponding to the three-dimensional Baxter-Bazhanov model. The vertex type
weight function is parametrized as the dihedral angles between the rapidity
planes connected with the cube. And we write down the symmetry relations of the
weight functions under the actions of the symmetry group $G$ of the cube. The
six angles with a constrained condition, appeared in the tetrahedron equation,
can be regarded as the six spectrums connected with the six spaces in which the
vertex type tetrahedron equation is defined.
