Limit set

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Three-generator quasi-fuchsian groups

fractal3D cover image

Quasi-fuchsian fractals had been studied as limit sets of two-generator quasi-fuchsian groups by the great German geometer, Felix Klein in nineteenth-century. Although very few examples of quais-fuchsian groups in three dimensional space are known, they had never been visualized. The limits set of fuchsian group can be defined as round sphere, and those of quasi-fuchsian group is just known as not round surfaces which topologically is equivalent to sphere, but no one knows how non-round they are.

In 2002, Yoshiaki Araki and Kazushi Ahara proposed a method to classify the family of quasi-fuchsian groups in three dimensional space and visualize the limit sets as 3D fractals with the power of computer graphics. You now can see how the sphere get non-round surface as fractals!

3D quasi-fuchsian fractals are the first ever waiting "native" 3D fractal shapes which have richer nature of fractals than other known 3D fractals. 3D quasi-fuchsian fractals have infinitely complicated surfaces and bended horns with the beauty of self-similarity. There are no smooth orbits which are found in quaternion fractals as a slice of fourth dimensional space onto three dimensional space. No straight lines founds like Menger's sponge as the part of edges of initial polyhedron.


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