Indra's Pearls: The Vision of Felix Klein
Cambridge University Press, Apr 25, 2002 - Mathematics - 395 pages
Felix Klein, a great geometer of the nineteenth century, rediscovered an idea from Hindu mythology in mathematics: the heaven of Indra in which the whole Universe was mirrored in each pearl in a net of pearls. Practically impossible to represent by hand, this idea barely existed outside the imagination, until the 1980s when the authors embarked on the first computer investigation of Klein's vision. In this extraordinary book they explore the path from some basic mathematical ideas to the simple algorithms that create delicate fractal filigrees, most appearing in print for the first time. Step-by-step instructions for writing computer programs allow beginners to generate the images.
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The language of symmetry
A delightful fiction
Double spirals and Mobius maps
The Schottky dance
Fractal dust and infinite words
The glowing gasket
Playing with parameters
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abAB algebra algorithm angle Apollonian gasket basic blue boundary calculate called centre chain Chapter coloured complex numbers conjugate corresponding curve cusp groups cyclic permutations depth-first search double cusp equation exactly example Farey Farey sequence fixed point formula fractal fractions frame of Figure Fuchsian geometry half plane Hausdorff dimension hyperbolic ideal triangle imaginary infinite words inside integer inverse Klein Kleinian groups labelled limit points limit set look loop loxodromic Maskit slice mathematical mathematician matrix Mobius maps Mobius transformations modular group move multiply nesting non-Euclidean ordinary set p/q word pair parabolic parameters pattern picture plot quasicircle radius real numbers region repetends Riemann sphere rotation Schottky circles Schottky disks Schottky group sequence shows shrink spiral square stereographic projection Stickler symmetry tangency points tangent tangent circles tile torus trace translation unit circle vertical word tree yellow
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