  Take $A$ to be a regular quadratic algebra of global dimension three. We
observe that there are examples of $A$ containing a dimension three regular
cubic algebra $C$. If $B$ is another dimension three regular quadratic algebra,
also containing $C$ as a subalgebra, then we can form the pushout algebra $D$
of the inclusions $i_1:C\hookrightarrow A$ and $i_2:C\hookrightarrow B$. We
show that for a certain class of regular algebras $C\hookrightarrow A,B$, their
pushouts $D$ are regular quadratic algebras of global dimension four.
Furthermore, some of the point module structures of the dimension three
algebras get passed on to the pushout algebra $D$.
