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By Katz N.M.

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35(3), 193–214 (1998) 37. : Teichmüller space and fundamental domains of Fuchsian groups. Enseign. Math. (2) 45(1-2), 169–187 (1999) 38. : Circle patterns with the combinatorics of the square grid. Duke Math. J. 86, 347–389 (1997) 39. : DGD Gallery, Lawson’s surface uniformization. de/models/lawsons_surface_uniformization (2015) 40. : Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001) 41. : Conformal invariance in random cluster models.

53 Fig. 34 Left A surface glued from six squares. Right Fuchsian uniformization and fundamental domain For each representation we choose corresponding fundamental polygons that allow the comparison of the uniformization: • an octagon with canonical edge pairing aba b cdc d , • an octagon with opposite sides identified, abcda b c d , • a 12-gon that is adapted to the six-squares surface. All data presented in this section is available on the DGD Gallery webpage [39]. Hyperelliptic curve. We uniformize the hyperelliptic curve μ2 = λ6 − 1 as described in Sect.

To achieve this we choose points with normally distributed coordinates and project them to S 2 [31]. 4 Numerical Experiments Given the branch points of an elliptic curve, the modulus τ can be calculated in terms of hypergeometric functions. In this section, we compare the theoretical value of τ with the value τˆ that we obtain by the discrete uniformization method explained in Sect. 2. I. Bobenko et al. so the branch points λ1 , λ2 , λ3 , ∞ satisfy λ1 + λ2 + λ3 = 0, and g2 = −4(λ1 λ2 + λ2 λ3 + λ3 λ1 ), g3 = 4λ1 λ2 λ3 .

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