  In the early 1990s Steve Gersten and Hamish Short proved that compact
nonpositively curved triangle complexes have biautomatic fundamental groups and
that compact nonpositively curved square complexes have biautomatic fundamental
groups. In this article we report on the extent to which results such as these
extend to nonpositively curved complexes built out a mixture of triangles and
squares. Since both results by Gersten and Short have been generalized to
higher dimensions, this can be viewed as a first step towards unifying
Januszkiewicz and {\'S}wi{\.a}tkowski's theory of simplicial nonpositive
curvature with the theory of nonpositively curved cube complexes.
