<s>
Fractal-generating	B-Algorithm
software	I-Algorithm
is	O
any	O
type	O
of	O
graphics	B-Application
software	I-Application
that	O
generates	O
images	O
of	O
fractals	B-Application
.	O
</s>
<s>
There	O
are	O
many	O
fractal	B-Application
generating	O
programs	O
available	O
,	O
both	O
free	O
and	O
commercial	O
.	O
</s>
<s>
Mobile	O
apps	O
are	O
available	O
to	O
play	O
or	O
tinker	O
with	O
fractals	B-Application
.	O
</s>
<s>
Some	O
programmers	O
create	O
fractal	B-Application
software	O
for	O
themselves	O
because	O
of	O
the	O
novelty	O
and	O
because	O
of	O
the	O
challenge	O
in	O
understanding	O
the	O
related	O
mathematics	O
.	O
</s>
<s>
The	O
generation	O
of	O
fractals	B-Application
has	O
led	O
to	O
some	O
very	O
large	O
problems	O
for	O
pure	O
mathematics	O
.	O
</s>
<s>
Fractal	B-Algorithm
generating	I-Algorithm
software	I-Algorithm
creates	O
mathematical	O
beauty	O
through	O
visualization	B-Application
.	O
</s>
<s>
Modern	O
computers	O
may	O
take	O
seconds	O
or	O
minutes	O
to	O
complete	O
a	O
single	O
high	B-Algorithm
resolution	I-Algorithm
fractal	B-Application
image	O
.	O
</s>
<s>
Images	O
are	O
generated	O
for	O
both	O
simulation	O
(	O
modeling	O
)	O
and	O
random	O
fractals	B-Application
for	O
art	O
.	O
</s>
<s>
Fractal	B-Application
generation	O
used	O
for	O
modeling	O
is	O
part	O
of	O
realism	O
in	O
computer	O
graphics	O
.	O
</s>
<s>
Fractal	B-Application
generation	O
software	O
can	O
be	O
used	O
to	O
mimic	O
natural	O
landscapes	O
with	O
fractal	B-Algorithm
landscapes	I-Algorithm
and	O
scenery	O
generation	O
programs	O
.	O
</s>
<s>
Fractal	B-Application
imagery	O
can	O
be	O
used	O
to	O
introduce	O
irregularity	O
to	O
an	O
otherwise	O
sterile	O
computer	O
generated	O
environment	O
.	O
</s>
<s>
Fractals	B-Application
are	O
generated	O
in	O
music	B-Application
visualization	I-Application
software	O
,	O
screensavers	B-General_Concept
and	O
wallpaper	O
generators	O
.	O
</s>
<s>
Because	O
complex	O
images	O
can	O
be	O
generated	O
from	O
simple	O
formula	O
fractals	B-Application
are	O
often	O
used	O
among	O
the	O
demoscene	O
.	O
</s>
<s>
The	O
generation	O
of	O
fractals	B-Application
such	O
as	O
the	O
Mandelbrot	O
set	O
is	O
time-consuming	O
and	O
requires	O
many	O
computations	O
,	O
so	O
it	O
is	O
often	O
used	O
in	O
benchmarking	O
devices	O
.	O
</s>
<s>
The	O
generation	O
of	O
fractals	B-Application
by	O
calculation	O
without	O
computer	O
assistance	O
was	O
undertaken	O
by	O
German	O
mathematician	O
Georg	O
Cantor	O
in	O
1883	O
to	O
create	O
the	O
Cantor	O
set	O
.	O
</s>
<s>
Throughout	O
the	O
following	O
years	O
,	O
mathematicians	O
have	O
postulated	O
the	O
existence	O
of	O
numerous	O
fractals	B-Application
.	O
</s>
<s>
Some	O
were	O
conceived	O
before	O
the	O
naming	O
of	O
fractals	B-Application
in	O
1975	O
,	O
for	O
example	O
,	O
the	O
Pythagoras	B-Algorithm
tree	I-Algorithm
by	O
Dutch	O
mathematics	O
teacher	O
Albert	O
E	O
.	O
Bosman	O
in	O
1942	O
.	O
</s>
<s>
The	O
development	O
of	O
the	O
first	O
fractal	B-Algorithm
generating	I-Algorithm
software	I-Algorithm
originated	O
in	O
Benoit	O
Mandelbrot	O
's	O
pursuit	O
of	O
a	O
generalized	O
function	O
for	O
a	O
class	O
of	O
shapes	O
known	O
as	O
Julia	B-Language
sets	I-Language
.	O
</s>
<s>
In	O
1979	O
,	O
Mandelbrot	O
discovered	O
that	O
one	O
image	O
of	O
the	O
complex	O
plane	O
could	O
be	O
created	O
by	O
iteration	B-Algorithm
.	O
</s>
<s>
He	O
and	O
programmers	O
working	O
at	O
IBM	O
generated	O
the	O
first	O
rudimentary	O
fractal	B-Application
printouts	O
.	O
</s>
<s>
This	O
marked	O
the	O
first	O
instance	O
of	O
the	O
generation	O
of	O
fractals	B-Application
by	O
non-linear	O
creations	O
laws	O
or	O
'	O
escape	O
time	O
fractal	B-Application
 '	O
.	O
</s>
<s>
The	O
October	O
1983	O
issue	O
of	O
Acorn	O
User	O
magazine	O
carried	O
a	O
BBC	B-Language
BASIC	I-Language
listing	O
for	O
generating	O
fractal	B-Application
shapes	O
by	O
Susan	O
Stepney	O
,	O
now	O
Professor	O
of	O
Computer	O
Science	O
at	O
the	O
University	O
of	O
York	O
.	O
</s>
<s>
She	O
followed	O
this	O
up	O
in	O
the	O
March	O
1984	O
Acorn	O
User	O
with	O
“	O
Snowflakes	O
and	O
other	O
fractal	B-Application
monsters	O
”	O
.	O
</s>
<s>
Fractals	B-Application
were	O
rendered	O
in	O
computer	O
games	O
as	O
early	O
as	O
1984	O
with	O
the	O
creation	O
of	O
Rescue	B-Application
on	I-Application
Fractalus	I-Application
!	I-Application
.	O
</s>
<s>
From	O
the	O
early	O
1980s	O
to	O
about	O
1995	O
hundreds	O
of	O
different	O
fractal	B-Application
types	O
were	O
formulated	O
.	O
</s>
<s>
The	O
generation	O
of	O
fractal	B-Application
images	O
grew	O
in	O
popularity	O
as	O
the	O
distribution	O
of	O
computers	O
with	O
a	O
maths	O
co-processor	O
or	O
floating-point	B-General_Concept
unit	I-General_Concept
in	O
the	O
central	O
processing	O
unit	O
were	O
adopted	O
throughout	O
the	O
1990s	O
.	O
</s>
<s>
At	O
this	O
time	O
the	O
rendering	O
of	O
high	B-Algorithm
resolution	I-Algorithm
VGA	O
standard	O
images	O
could	O
take	O
many	O
hours	O
.	O
</s>
<s>
Fractal	B-Application
generation	O
algorithms	O
display	O
extreme	O
parallelizability	O
.	O
</s>
<s>
Fractal-generating	B-Algorithm
software	I-Algorithm
was	O
rewritten	O
to	O
make	O
use	O
of	O
multi-threaded	B-Architecture
processing	I-Architecture
.	O
</s>
<s>
Subsequently	O
,	O
the	O
adoption	O
of	O
graphics	B-Architecture
processing	I-Architecture
units	I-Architecture
in	O
computers	O
has	O
greatly	O
increased	O
the	O
speed	O
of	O
rendering	O
and	O
allowed	O
for	O
real-time	B-General_Concept
changes	I-General_Concept
to	O
parameters	O
that	O
were	O
previously	O
impossible	O
due	O
to	O
render	O
delay	O
.	O
</s>
<s>
3D	O
fractal	B-Application
generation	O
emerged	O
around	O
2009	O
.	O
</s>
<s>
An	O
early	O
list	O
of	O
fractal-generating	B-Algorithm
software	I-Algorithm
was	O
compiled	O
for	O
the	O
book	O
titled	O
Fractals	B-Application
:	O
The	O
Patterns	O
of	O
Chaos	O
by	O
John	O
Briggs	O
published	O
in	O
1992	O
.	O
</s>
<s>
There	O
are	O
two	O
major	O
methods	O
of	O
two	O
dimensional	O
fractal	B-Application
generation	O
.	O
</s>
<s>
One	O
is	O
to	O
apply	O
an	O
iterative	B-Algorithm
process	O
to	O
simple	O
equations	O
by	O
generative	O
recursion	O
.	O
</s>
<s>
In	O
fractal	B-Application
software	O
values	O
for	O
a	O
set	O
of	O
points	O
on	O
the	O
complex	O
plane	O
are	O
calculated	O
and	O
then	O
rendered	O
as	O
pixels	B-Algorithm
.	O
</s>
<s>
This	O
computer-based	O
generation	O
of	O
fractal	B-Application
objects	O
is	O
an	O
endless	O
process	O
.	O
</s>
<s>
In	O
theory	O
,	O
images	O
can	O
be	O
calculated	O
infinitely	B-Application
but	O
in	O
practice	O
are	O
approximated	O
to	O
a	O
certain	O
level	O
of	O
detail	O
.	O
</s>
<s>
The	O
maximum	O
fractal	B-Application
dimension	O
that	O
can	O
be	O
produced	O
varies	O
according	O
to	O
type	O
and	O
is	O
sometimes	O
limited	O
according	O
to	O
the	O
method	O
implemented	O
.	O
</s>
<s>
Colour	B-Algorithm
banding	I-Algorithm
may	O
appear	O
in	O
images	O
depending	O
on	O
the	O
method	O
of	O
coloring	O
used	O
as	O
well	O
as	O
gradient	O
color	O
density	O
.	O
</s>
<s>
Some	O
programs	O
generate	O
geometric	O
self-similar	O
or	O
deterministic	O
fractals	B-Application
such	O
as	O
the	O
Koch	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
These	O
simple	O
fractals	B-Application
originate	O
from	O
a	O
technique	O
first	O
proposed	O
in	O
1904	O
by	O
Koch	O
.	O
</s>
<s>
The	O
other	O
main	O
method	O
is	O
with	O
Iterated	B-Algorithm
Function	I-Algorithm
Systems	I-Algorithm
consisting	O
of	O
a	O
number	O
of	O
affine	O
transformations	O
.	O
</s>
<s>
In	O
the	O
first	O
method	O
each	O
pixel	B-Algorithm
in	O
a	O
fractal	B-Application
image	O
is	O
evaluated	O
according	O
to	O
a	O
function	O
and	O
then	O
coloured	O
,	O
before	O
the	O
same	O
process	O
is	O
applied	O
to	O
the	O
next	O
pixel	B-Algorithm
.	O
</s>
<s>
The	O
former	O
method	O
represents	O
the	O
classical	O
stochastic	O
approach	O
while	O
the	O
latter	O
implements	O
a	O
linear	O
fractal	B-Application
model	O
.	O
</s>
<s>
Three	O
dimensional	O
fractals	B-Application
are	O
generated	O
in	O
a	O
variety	O
of	O
ways	O
including	O
by	O
using	O
quaternion	O
algebra	O
.	O
</s>
<s>
Fractals	B-Application
emerge	O
from	O
fluid	O
dynamics	O
modelling	O
simulations	O
as	O
turbulence	O
when	O
contour	O
advection	O
is	O
used	O
to	O
study	O
chaotic	O
mixing	O
.	O
</s>
<s>
The	O
Buddhabrot	B-Algorithm
method	O
was	O
introduced	O
in	O
1993	O
.	O
</s>
<s>
Programs	O
might	O
use	O
fractal	B-Application
heightmaps	B-Data_Structure
to	O
generate	O
terrain	O
.	O
</s>
<s>
Fractals	B-Application
have	O
been	O
generated	O
on	O
computers	O
using	O
the	O
following	O
methods	O
:	O
Menger	B-Algorithm
sponge	I-Algorithm
,	O
Hypercomplex	O
manifold	O
,	O
Brownian	O
tree	O
,	O
Brownian	O
motion	O
,	O
Decomposition	B-Application
,	O
L-systems	B-Language
,	O
Lyapunov	B-Algorithm
fractals	I-Algorithm
,	O
Newton	B-Algorithm
fractals	I-Algorithm
,	O
Pickover	B-Algorithm
stalks	I-Algorithm
and	O
Strange	O
attractors	O
.	O
</s>
<s>
Many	O
different	O
features	O
are	O
included	O
in	O
fractal-generating	B-Algorithm
software	I-Algorithm
packages	O
.	O
</s>
<s>
Most	O
feature	O
some	O
form	O
of	O
algorithm	O
selection	O
,	O
an	O
interactive	O
image	B-Application
zoom	I-Application
,	O
and	O
the	O
ability	O
to	O
save	O
files	O
in	O
JPEG	O
,	O
TIFF	O
,	O
or	O
PNG	O
format	O
,	O
as	O
well	O
as	O
the	O
ability	O
to	O
save	O
parameter	O
files	O
,	O
allowing	O
the	O
user	O
to	O
easily	O
return	O
to	O
previously	O
created	O
images	O
for	O
later	O
modification	O
or	O
exploration	O
.	O
</s>
<s>
The	O
formula	O
,	O
parameters	O
,	O
variables	O
and	O
coloring	O
algorithms	O
for	O
fractal	B-Application
images	O
can	O
be	O
exchanged	O
between	O
users	O
of	O
the	O
same	O
program	O
.	O
</s>
<s>
There	O
is	O
no	O
universally	O
adopted	O
standard	O
fractal	B-Application
file	O
format	O
.	O
</s>
<s>
One	O
feature	O
of	O
most	O
escape	O
time	O
fractal	B-Application
programs	O
or	O
algebraic-based	O
fractals	B-Application
is	O
a	O
maximum	O
iteration	B-Algorithm
setting	O
.	O
</s>
<s>
Increasing	O
the	O
iteration	B-Algorithm
count	O
is	O
required	O
if	O
the	O
image	O
is	O
magnified	O
so	O
that	O
fine	O
detail	O
is	O
not	O
lost	O
.	O
</s>
<s>
Limiting	O
the	O
maximum	O
iterations	B-Algorithm
is	O
important	O
when	O
a	O
device	O
's	O
processing	O
power	O
is	O
low	O
.	O
</s>
<s>
Because	O
of	O
the	O
convenient	O
ability	O
to	O
add	O
post-processing	O
effects	O
layering	O
and	O
alpha	B-Algorithm
compositing	I-Algorithm
features	O
found	O
in	O
other	O
graphics	B-Application
software	I-Application
have	O
been	O
included	O
.	O
</s>
<s>
Many	O
packages	O
also	O
allow	O
the	O
user	O
to	O
input	O
their	O
own	O
formula	O
,	O
to	O
allow	O
for	O
greater	O
control	O
of	O
the	O
fractals	B-Application
,	O
as	O
well	O
as	O
a	O
choice	O
of	O
color	O
rendering	O
,	O
along	O
with	O
the	O
use	O
of	O
filters	O
and	O
other	O
image	O
manipulation	O
techniques	O
.	O
</s>
<s>
Some	O
fractal	B-Application
software	O
packages	O
allow	O
for	O
the	O
creation	O
of	O
movies	O
from	O
a	O
sequence	O
of	O
fractal	B-Application
images	O
.	O
</s>
<s>
Others	O
display	O
render	O
time	O
and	O
allow	O
some	O
form	O
of	O
color	B-Device
cycling	I-Device
and	O
color	B-Data_Structure
palette	I-Data_Structure
creation	O
tools	O
.	O
</s>
<s>
Standard	O
graphics	B-Application
software	I-Application
(	O
such	O
as	O
GIMP	B-Application
)	O
contains	O
filters	O
or	O
plug-ins	O
which	O
can	O
be	O
used	O
for	O
fractal	B-Application
generation	O
.	O
</s>
<s>
Blender	B-Application
contains	O
a	O
fractal	B-Application
(	O
or	O
random	O
)	O
modifier	O
.	O
</s>
<s>
Many	O
stand-alone	O
fractal-generating	O
programs	O
can	O
be	O
used	O
in	O
conjunction	O
with	O
other	O
graphics	B-Application
programs	I-Application
(	O
such	O
as	O
Photoshop	B-Operating_System
)	O
to	O
create	O
more	O
complex	O
images	O
.	O
</s>
<s>
POV-Ray	B-Application
is	O
a	O
ray	O
tracing	O
program	O
which	O
generates	O
images	O
from	O
a	O
text-based	O
scene	O
description	O
that	O
can	O
generate	O
fractals	B-Application
.	O
</s>
<s>
Scripts	O
on	O
3ds	B-Language
Max	I-Language
and	O
Autodesk	B-Language
Maya	I-Language
can	O
be	O
used	O
.	O
</s>
<s>
A	O
number	O
of	O
web-based	O
interfaces	O
for	O
the	O
fractal	B-Application
generation	O
are	O
freely	O
available	O
including	O
Turtle	O
Graphics	O
Renderer	O
.	O
</s>
<s>
Fractal	B-Application
Lab	O
can	O
generate	O
both	O
2D	O
and	O
3D	O
fractals	B-Application
and	O
is	O
available	O
over	O
the	O
web	O
using	O
WebGL	B-Library
.	O
</s>
<s>
JWildfire	O
is	O
a	O
java-based	O
,	O
open-source	B-Application
fractal	B-Algorithm
flame	I-Algorithm
generator	O
.	O
</s>
<s>
Mandelbrot	O
Fractal	B-Application
is	O
a	O
fractal	B-Application
explorer	O
written	O
in	O
JavaScript	B-Language
.	O
</s>
<s>
Fractal	B-Application
Grower	O
is	O
software	O
written	O
in	O
Java	O
for	O
generating	O
Lindenmayer	O
Substitution	O
Fractals	B-Application
(	O
L-systems	B-Language
)	O
.	O
</s>
<s>
Because	O
of	O
the	O
butterfly	O
effect	O
,	O
generating	O
fractals	B-Application
can	O
be	O
difficult	O
to	O
master	O
.	O
</s>
<s>
This	O
includes	O
the	O
characteristics	O
of	O
fractal	B-Application
dimension	O
,	O
recursion	O
and	O
self-similarity	O
exhibited	O
by	O
all	O
fractals	B-Application
.	O
</s>
<s>
There	O
are	O
many	O
fractal	B-Application
generating	O
programs	O
available	O
,	O
both	O
free	O
and	O
commercial	O
.	O
</s>
<s>
Notable	O
fractal	B-Application
generating	O
programs	O
include	O
:	O
</s>
<s>
Chaotica	B-Algorithm
–	O
commercial	O
IFS	O
software	O
for	O
Windows	O
,	O
Linux	B-Application
and	O
Mac	B-Application
OS	I-Application
.	O
</s>
<s>
Electric	B-Algorithm
Sheep	I-Algorithm
–	O
open	B-Application
source	I-Application
distributed	O
screensaver	B-General_Concept
software	O
,	O
developed	O
by	O
Scott	O
Draves	O
.	O
</s>
<s>
Most	O
of	O
the	O
above	O
programs	O
make	O
two-dimensional	O
fractals	B-Application
,	O
with	O
a	O
few	O
creating	O
three-dimensional	O
fractal	B-Application
objects	O
,	O
such	O
as	O
mandelbulbs	B-Algorithm
and	O
mandelboxes	B-Algorithm
.	O
</s>
<s>
Mandelbulber	O
is	O
an	O
experimental	O
,	O
cross	O
platform	O
open-source	B-Application
program	I-Application
that	O
generates	O
three-dimensional	O
fractal	B-Application
images	O
.	O
</s>
<s>
Mandelbulb	B-Algorithm
3D	O
is	O
free	O
software	O
for	O
creating	O
3D	O
images	O
featuring	O
many	O
effects	O
found	O
in	O
3D	O
rendering	O
environments	O
.	O
</s>
<s>
Incendia	O
is	O
a	O
3D	O
fractal	B-Application
program	O
that	O
uses	O
Iterated	B-Algorithm
Function	I-Algorithm
Systems	I-Algorithm
(	O
IFS	O
)	O
for	O
fractal	B-Application
generation	O
.	O
</s>
<s>
The	O
open	B-Application
source	I-Application
GnoFract	O
4D	O
is	O
available	O
.	O
</s>
<s>
ChaosPro	O
is	O
freeware	B-License
fractal	B-Application
creation	O
program	O
.	O
</s>
<s>
Fraqtive	O
is	O
an	O
open	B-Application
source	I-Application
cross	O
platform	O
fractal	B-Application
generator	O
.	O
</s>
<s>
MandelX	O
is	O
a	O
free	O
program	O
for	O
rendering	O
fractal	B-Application
images	O
on	O
Windows	O
.	O
</s>
<s>
WinCIG	O
,	O
Chaoscope	O
,	O
Tierazon	O
,	O
Fractal	B-Application
Forge	O
and	O
Malsys	O
also	O
generate	O
fractal	B-Application
images	O
.	O
</s>
