A non-traditional approach to the discretization of differential-geometrical connections was suggested by the authors [Russ. Math. Surv. 52, No. 6, 1294--1295 (1997); translation from Usp. Mat. Nauk 52, No. 6, 157--158 (1997; Zbl 0928.39009
)]. At the same time, they started with first-order difference “black-and-white triangle operators (equations)” on triangulated surfaces with a black-and-white coloring of triangles. In the present work, they develop a theory of these operators and equations showing their similarity to the complex derivatives $\partial$ and $-\partial$.