2004/11/20 | Klein四次曲线[Klein Quartic]
类别(∑〖数学〗) | 评论(0) | 阅读(56) | 发表于 16:00

A surface with equation

in complex projective coordinates (Levy 1999, p. ix; left figure), and with the ideal surface determined by the equation

(Thurston 1999, p. 3; right figure), also known as the Klein curve.

Klein (1859; translation reprinted in 1999) discovered that this surface has a number of remarkable properties, including an incredible 336-fold symmetry when mirror reflections are allowed (Levy 1999, p. ix; Thurston 1999, p. 2), a number that later was found to be the maximum possible for a curve of its type (Hurwitz 1893; Karcher and Weber 1999, p. 9). Klein arrives at this equation as a quotient of the upper half-plane by the modular group of fractional linear transformations whose coefficients are integers and that reduce to the identity modulo 7 (Levy 1999, p. ix).

The abstract surface cannot be rendered exactly in three-dimensional space, but topologically, the Klein quartic is a three-holed torus (Thurston 1999, pp. 1 and 4). The Klein quartic can be viewed as an extension of the concept of the Platonic solids to a hyperbolic heptagonal tiling, as illustrated above (Coxeter 1954; Thurston 1999, p. 7; Wolfram 2002, p. 1050). In the tiling, the number of heptagons in the nth "ring" is, amazingly, equal to , where is a Fibonacci number (Thurston 1999, p. 5).

The surface has be sculpted by Helaman Ferguson in marble and serpentine, and was unveiled at the Mathematical Sciences Research Institute in Berkeley on November 14, 1993 (Ley 1999, Plate 1 following p. 142; Borwein and Bailey 2003, p. 55, color plate IV, and back cover).
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