Maybe the easiest way to visualize how a dessin (a bicolored graph embedded in a surface) classifies Riemann surfaces is to sketch a flow field where, say, the black vertices are sources and the white vertices are sinks. Notice in the sketch above that the flow lines only reach nearest neighbors bordering a given face. Seeing the sketched flow lines as lines of longitude on the Earth (say from North to South) or as lines of constant phase angle on the Riemann sphere (from 0 to infinity) suggests a particular way in which multiple instances of the Riemann sphere can cover the surface.
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