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

Triangles, squares and geodesics