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Hexagonal an triangular network by a second degree polynomial

In the realm of discrete analytic functions theory, a certain way to look at discrete holomorphic function arises with the help of the primal and dual graphs of a cellular decomposition of a surface. For example the triangular and hexagonal lattices in the plane. In degree 2, the discrete polynomials agree with the usual polynomials so it is easy to explore this space.
You can play with the values of the polynomial . Since it is of degree two, only very special values of the parameters will put one on top of the other! First of all you realise that the translation factor is of no interest, it just translates everything so let's assume . Then is just a scaling factor, so you can put it to 1, therefore the only free parameter is . Try . This applet was developed in order to fuel a discussion with Araki Yoshiaki.