On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line is a classical object and the curves are quantum. This generalizes to a new connection between algebraic geometry and the geometry of polyhedra, which is more straight-forward than the other known connections and gives a new insight into constructions used in the topology of real algebraic varieties.

12 pages, 3 figures, Plenary talk at the 3rd ECM, Barcelona, July 10-14, 2000. Sections 2.2, 3.3, 3.4 changed, 2.3 removed to correct consequences of a miscalculation, a reference updated