# Fractals/Mathematics/group/Kleinian group

Kleinian group

## algorithms

* escape time by Jos Leys, FF discussion
* "drawn up by a multiplication of two by two matrices representing Moebius transformation"
* inverse algorithm
* Drawing limit sets of Kleinian groups using finite state automata by Greg McShane, John R. Parker, Ian Redfern
* Kleinian circle packings by John R. Parker
* An index-based revisitation by Alessandro Rosa

## Programs

* Spirofractal
* xenodream - Unique 3D graphics software gives you results that would be difficult or impossible using other tools, and more fun doing it.
* kleinian by Danny Calegari
* Kleinian Groups Programs: Indra Version by D Wright
* lim by Curtis McMullen
* Kleinian and Quasi-Fuchsian Limit Sets: An Open Source Toolbox Genotype: 1.1.45 by Chris King
* Arnaud Cheritat
** Universal cover of Klein's quartic. and the code
* shadertoy :
** kleinian
* soma-arc
** soma-arc : SchottkyLink
** A New Algorithm for Rendering Kissing Schottky Groups by Kento Nakamura, and Kazushi Ahara
** KleinianWalkerWeb
* webgl
** Universal cover of Klein's quartic. CC-BY-SA Arnaud Chéritat
** aplets by A.N.Anderson
** CindyJS
** Circles by Alessandro Rosa
* maple
** Yousi Daniel Hue
* JS
** Draw the limit set of a group of Möbius transformations by Tim Hutton

## relation between kleinian group and fractals

* "for Fuchsian groups of the first kind, the limit set has Hausdorff dimension 1, i.e., it is not fractal.
* ... for all other quasifuchsian groups of the first kind, the limit set has Hausdorff dimension strictly bigger than 1, by a theorem of Rufus Bowen."
* Schottky group

## Video

* vimeo album Non-Euclidean Tessellations Tessellations of Kleinian Groups.
* Indra's Pearls fractal animation by FreymanArt
* Indra's Pearls: A Mathematical Adventure
* youtube search kleinian
** An overview on complex Kleinian groups (Angel Cano)
* Schottky Link - Introduction by soma
* KleinianWalker - Introduction by soma

## images

* by Jos Lays
* by Curtis McMullen
* True 3d kleinians can be produced with surfbox by pupukuusikko
* Kleinian and Quasi-Fuchsian Limit Sets: An Open Source Toolbox Genotype: 1.1.32 by Chris King
* Generation of Limit Sets of Kleinian Groups Using a Feedback Loop Approach - java screipt
* David Dumas
** SL(View) 0.4 by David Dumas
* Limit set of a quasifuchsian group on the sphere
* Schottky (Kleinian) group limit set in the plane

# References

* ↑ Kleinian group in wikipedia
* ↑ A fast algorithm for limit sets of Kleinian groups with the Maskit parametrisation. Jos Leys January 2017
* ↑ hiddendimension : KleinianGroup_Fractals
* ↑ fractalforums.org : kleinian-adventures-three-generator-maskit
* ↑ http://www.alunw.freeuk.com/fractalgallery.html
* ↑ openprocessing code
* ↑ GLUT Kleinian group visualizer by Danny Calegari
* ↑ kleinian, a tool for visualizing Kleinian groups Posted on March 4, 2014 by Danny Calegari
* ↑ kleinian by D Wright
* ↑ FRactal forum : Kleinian groups - an immense collection with sources!
* ↑ Programs by Curtis McMullen
* ↑ Kleinian and Quasi-Fuchsian Limit Sets: An Open Source Toolbox Genotype: 1.1.45 by Chris King
* ↑ mathoverflow question: why-are-fuchsian-groups-interesting
* ↑ The Fractal Nature of Schottky Groups John Conley May 31, 2013
* ↑ A New Algorithm for Rendering Kissing Schottky Groups by Kento Nakamura, and Kazushi Ahara
* ↑ Jos Leys kleinian galleries
* Klein's Web Pages
** Indra's Pearls Web Site
** Non-Euclidean geometry and Indra's pearls By Caroline Series and David Wright
** FF
* The Farey-Ford tessellation and circle packing
* Chapter : Klienian goups in book Geometry and topology of three-manifolds By William Thurston
* The Group Properties Wiki
* wiki
* algorithms by A N Anderson
* Interaktive Materialien zum Buch Indra's Pearls by StefanLembach - 12 May 2010
* COMPLEX KLEINIAN GROUPS by Angel Cano , Juan-Pablo Navarrete, Jos´e Seade
* FF : kleinian-adventures-three-generator-maskit by xenodreambuie
* CREATING SOFTWARE FOR VISUALIZING KLEINIAN GROUPS by Yasushi Yamashita, in Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore: Volume 23
Geometry, Topology and Dynamics of Character Varieties https://doi.org/10.1142/8445 | August 2012
* fractalforums.org : help-with-indras-pearls-by-david-mumford-caroline-series-and-david-wright