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In this century, a square-tiled translation surface (an origami) isintensively studied as an object with special properties of its translationstructure and its $SL(2,\mathbb{R})$-orbit embedded in the moduli space. Wegeneralize this concept in the language of flat surfaces appearing naturally inthe Teichm\"uller theory. We study the combinatorial structure of origamis andshow that a certain system of linear equations realizes the flat surface inwhich rectangles of specified moduli replace squares of an origami. Thisconstruction gives a parametrization of the family of flat surfaces with twofinite Jenkins-Strebel directions for each combinatorial structure oftwo-directional cylinder decomposition. Moreover, we obtain the inclusion ofVeech groups of such flat surfaces under a covering relation with specificbranching behavior.

General origamis and Veech groups of flat surfaces