0 Conformally symmetric circle packings . A generalization of Doyle spirals
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چکیده
Circle packings (and more generally patterns) as discrete analogs of conformal mappings is a fast developing field of research on the border of analysis and geometry. Recent progress was initiated by Thurston’s idea [T] about the approximation of the Riemann mapping by circle packings. The corresponding convergence was proven by Rodin and Sullivan [RS]; many additional connections with analytic functions, such as the discrete maximum principle and Schwarz’s lemma [R], the discrete uniformization theorem [BS], etc., have emerged since then. The topic “circle packings” is also a natural one for computer experimentation and visualization. Computer experiments demonstrate a surprisingly close analogy of the classical theory in the emerging “discrete analytic function theory” [DS]. Although computer experiments give convincing evidence for the existence of discrete analogs of many standard holomorphic functions, the Doyle spirals (which are discrete analogs of the exponential function, see section 4) are the only circle packings described explicitly. Circle packings are usually described analytically in the Euclidean setting, i.e. through their radii function. On the other hand, circles and the tangencies are preserved by the fractional-linear transformations of the Riemann sphere (Möbius transformations). It is natural to study circle packings in this setting, i.e. modulo the group of the Möbius transformations. He and Schramm [HS] developed a conformal description of hexagonal circle packings, which helped them to show that Thurston’s convergence of hexagonal circle packings to the Riemann mapping is actually C. They describe circle packings in terms of the cross-ratios
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تاریخ انتشار 2000