A Fractal Field Guide
This is a collection of limit sets rendered mostly between Jan'93 and Apr'94 using
a program that I was developing with the working title Live Fractal.
These pictures differ from the usual fractals you see in a few regards
 the defining mathematical formula come from families much more general than
the ones for Julia and Mandelbrot sets
 the pictures are rendered using inverse iteration as opposed to escape time
algorithms. As a result, only the limit sets themselves are shown, without
the razzle dazzle of a surrounding color field.
 the inverse iteration algorithm is able to handle not only the usual
IFS and Julia fractals, but also the limit sets of Kleinian groups, as
popularized in the book
Indra's Pearls.
In the list below, the accompanying descriptions are probably
vague and unhelpful, but should give an impression of the wide variety of
fractal types possible.
The rendering method used is in spirit the same as that popularized by Barnsley
for his IFS fractals. Many of the images have color which indicates the
number of times the algorithm visited a given region. Black indicates a
few hits, green some more, and red and yellow indicate many hits. Like photography,
the images you get depend on exposure time and filters. The series of images
presented here are more like polaroid snapshots than serious attempts at
creating high quality pictures.
Quadratic Rationals (including matings)

 inverse quadratic

 inverse quadratic

 a quadratic mating (rabbit+peapod)

 a quadratic mating (rabbit+basilica)

 a quadratic selfmating (peadpod+peapod)

 a quadratic mating

 quadratic self mating

 inverse quadratic

 quadratic rational

 quadratic rational

 quadratic rational

 quadratic rational

 inverse quadratic

 inverse quadratic
Carpet Fractals

 cubic version of a carpet fractal

 a carpet fractal

 a carpet fractal

 almost a carpet fractal

 a closeup of RABBIT

 a carpet fractal closeup
Conformal Maps and Correspondences

 derived from Barnsley's foggy coastline

 generated by Gauss's ArithmeticGeometric mean algorithm

 generated by Gauss's ArithmeticGeometric mean algorithm

 a quadratic correspondence

 a quadratic correspondence

 a perturbation of a mapping whose symmetry group is a dihedral group

 two superimposed members of a circle inversion group

 group of Moebius transformations

 group of Moebius transformations

 a quadratic correspondence

 not a fractal, but the image of a grid under an iterated conformal map

 see CONFORM2

 not a fractal, but the image of a circle under an iterated conformal map
Kleinian Groups

 a Kleinian group whose limit set consists of circles

 a Schottky group

 a Schottky group (verging on quasifuchsian)

 a Schottky group

 a Schottky group, showing the 4 generating circles

 a Kleinian group

 a nearly quasifuchsian group

 a quasifuchsian group

 partial rendering of a quasifuchsian group limit set

 10. quasifuchsian limit set  this is actually a nearly planefilling curve. This picture was used in the book
The Honors Class, by Ben Yandell

 11. A zoomed out version of 10. The box shows where 10 is situated.

 12. Same as 11, shows isometric circles

 13. a zoomed out version of 11. Note that peripheral resolution degrades.

 14. Zoomed out 13

 the limit set in 1014 rendered as a curve.

 the limit set in 1014 rendered as a curve (zoomed out)

 the limit set in 1014 rendered as a curve(zoomed out even more)

 in the neighbourhood of the Riley slice of Schottky space

 a Schottky group with 4 generating circles shown

 an outlying Schottky group
NonConformal Maps

 a nonconformal function

 a conjugate quadratic

 a nonconformal function

 a nonconformal function

 a nonconformal map (verging on strange attractor)
Glynn Sets
Quadratic and Cubic Julias

 a quadratic Julia set

 a dendritic rabbit

 a cubic Julia set

 a cubic Julia set

 a cubic Julia set

 a cubic Julia set

 a cubic Julia

 a cubic Julia

 a quadratic Julia

 a quadratic Julia set

 a quadratic Julia set (same as JULIA2, but with a white background)

 Julia set for z^10+c
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