Threegenerator quasifuchsian groups
Quasifuchsian fractals had been studied as limit sets of twogenerator quasifuchsian groups
by the great German geometer, Felix Klein in nineteenthcentury.
Although very few examples of quaisfuchsian 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 quasifuchsian group is just known as not round surfaces which topologically is equivalent to sphere, but no one knows how nonround they are.
In 2002, Yoshiaki Araki and Kazushi Ahara proposed a method to classify the family of
quasifuchsian 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 nonround surface as fractals!
3D quasifuchsian fractals are the first ever waiting "native" 3D fractal shapes which have richer nature of fractals
than other known 3D fractals.
3D quasifuchsian fractals have infinitely complicated surfaces and bended horns with the beauty of selfsimilarity.
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.
