xx Introduction integrable equations zero-curvature representation transformations B¨acklund- Darboux hierarchies of commuting flows ⇐= ⇐= ⇐= surfaces transformations surfaces and their multidimensional consistency Integrability Discrete Differential Geometry discrete nets ⇑ Differential Geometry ⇓ =⇒ =⇒ =⇒ Bianchi permutability CONSISTENCY Figure 0.2. The consistency principle of discrete differential geometry as conceptual basis of the differential geometry of special surfaces and of integrability. realized by a circle packing. According to Rodin-Sullivan (1987) the confor- mal Riemann map can be approximated by such circle packings (even with all the derivatives as shown by He-Schramm (1998)). The first convergence results concerning the transition from the middle to the left column in Figure 0.2 (from discrete to smooth differential geom- etry) were proven in Bobenko-Matthes-Suris (2003, 2005). This turns the general philosophy of discrete differential geometry into a firmly established

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