xviii Introduction discrete nets, is seen to rely on some incidence theorems of elementary ge- ometry. Conceptually, one can think of passing to a continuous limit by refining the mesh size in some of the lattice directions. In these directions the net converges to smooth surfaces whereas those directions that remain discrete correspond to transformations of the surfaces (see Figure 0.1). Differential geometric properties of special classes of surfaces and their transformations arise in this way from (and find their simple explanation in) the elemen- tary geometric properties of the original multidimensional discrete nets. In particular, diﬃcult classical theorems about the permutability of B¨acklund- Darboux type transformations (Bianchi permutability) for various geome- tries follow directly from the symmetry of the underlying discrete nets, and are therefore built in to the very core of the theory. Thus the transition from differential geometry to elementary geometry via discretization (or, in the opposite direction, the derivation of differential geometry from the discrete differential geometry) leads to enormous conceptual simplifications, and the true roots of the classical theory of special classes of surfaces are found in various incidence theorems of elementary geometry. In the classical differ- ential geometry these elementary roots remain hidden. The limiting process taking the discrete master theory to the classical one is inevitably accompa- nied by a break of the symmetry among the lattice directions, which always leads to structural complications. Figure 0.1. From the discrete master theory to the classical theory: surfaces and their transformations appear by refining two of three net directions. Finding simple discrete explanations for complicated differential-geomet- ric theories is not the only outcome of this development. It is well known that differential equations which analytically describe interesting special classes of surfaces are integrable (in the sense of the theory of integrable systems),

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