Three-generator quasi-fuchsian groups
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.