<s>
In	O
mathematics	O
,	O
iterated	B-Algorithm
function	I-Algorithm
systems	I-Algorithm
(	O
IFSs	O
)	O
are	O
a	O
method	O
of	O
constructing	O
fractals	B-Application
;	O
the	O
resulting	O
fractals	B-Application
are	O
often	O
self-similar	O
.	O
</s>
<s>
IFS	O
fractals	B-Application
are	O
more	O
related	O
to	O
set	O
theory	O
than	O
fractal	B-Application
geometry	O
.	O
</s>
<s>
IFS	O
fractals	B-Application
,	O
as	O
they	O
are	O
normally	O
called	O
,	O
can	O
be	O
of	O
any	O
number	O
of	O
dimensions	O
,	O
but	O
are	O
commonly	O
computed	O
and	O
drawn	O
in	O
2D	O
.	O
</s>
<s>
The	O
fractal	B-Application
is	O
made	O
up	O
of	O
the	O
union	O
of	O
several	O
copies	O
of	O
itself	O
,	O
each	O
copy	O
being	O
transformed	O
by	O
a	O
function	O
(	O
hence	O
"	O
function	O
system	O
"	O
)	O
.	O
</s>
<s>
Hence	O
,	O
the	O
shape	O
of	O
an	O
IFS	O
fractal	B-Application
is	O
made	O
up	O
of	O
several	O
possibly-overlapping	O
smaller	O
copies	O
of	O
itself	O
,	O
each	O
of	O
which	O
is	O
also	O
made	O
up	O
of	O
copies	O
of	O
itself	O
,	O
ad	O
infinitum	O
.	O
</s>
<s>
This	O
is	O
the	O
source	O
of	O
its	O
self-similar	O
fractal	B-Application
nature	O
.	O
</s>
<s>
Formally	O
,	O
an	O
iterated	B-Algorithm
function	I-Algorithm
system	I-Algorithm
is	O
a	O
finite	O
set	O
of	O
contraction	O
mappings	O
on	O
a	O
complete	O
metric	O
space	O
.	O
</s>
<s>
is	O
an	O
iterated	B-Algorithm
function	I-Algorithm
system	I-Algorithm
if	O
each	O
is	O
a	O
contraction	O
on	O
the	O
complete	O
metric	O
space	O
.	O
</s>
<s>
Random	O
elements	O
arbitrarily	O
close	O
to	O
S	O
may	O
be	O
obtained	O
by	O
the	O
"	O
chaos	B-Algorithm
game	I-Algorithm
,	O
"	O
described	O
below	O
.	O
</s>
<s>
The	O
collection	O
of	O
functions	O
generates	O
a	O
monoid	O
under	O
composition	B-Application
.	O
</s>
<s>
In	O
general	O
,	O
if	O
there	O
are	O
k	O
functions	O
,	O
then	O
one	O
may	O
visualize	O
the	O
monoid	O
as	O
a	O
full	O
k-ary	B-Data_Structure
tree	I-Data_Structure
,	O
also	O
known	O
as	O
a	O
Cayley	B-Algorithm
tree	I-Algorithm
.	O
</s>
<s>
Sometimes	O
each	O
function	O
is	O
required	O
to	O
be	O
a	O
linear	B-Architecture
,	O
or	O
more	O
generally	O
an	O
affine	B-Algorithm
,	O
transformation	O
,	O
and	O
hence	O
represented	O
by	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
However	O
,	O
IFSs	O
may	O
also	O
be	O
built	O
from	O
non-linear	O
functions	O
,	O
including	O
projective	B-Algorithm
transformations	I-Algorithm
and	O
Möbius	O
transformations	O
.	O
</s>
<s>
The	O
Fractal	B-Algorithm
flame	I-Algorithm
is	O
an	O
example	O
of	O
an	O
IFS	O
with	O
nonlinear	O
functions	O
.	O
</s>
<s>
The	O
most	O
common	O
algorithm	O
to	O
compute	O
IFS	O
fractals	B-Application
is	O
called	O
the	O
"	O
chaos	B-Algorithm
game	I-Algorithm
"	O
.	O
</s>
<s>
Each	O
of	O
these	O
algorithms	O
provides	O
a	O
global	O
construction	O
which	O
generates	O
points	O
distributed	O
across	O
the	O
whole	O
fractal	B-Application
.	O
</s>
<s>
If	O
a	O
small	O
area	O
of	O
the	O
fractal	B-Application
is	O
being	O
drawn	O
,	O
many	O
of	O
these	O
points	O
will	O
fall	O
outside	O
of	O
the	O
screen	O
boundaries	O
.	O
</s>
<s>
PIFS	O
(	O
partitioned	O
iterated	B-Algorithm
function	I-Algorithm
systems	I-Algorithm
)	O
,	O
also	O
called	O
local	O
iterated	B-Algorithm
function	I-Algorithm
systems	I-Algorithm
,	O
give	O
surprisingly	O
good	O
image	O
compression	O
,	O
even	O
for	O
photographs	O
that	O
do	O
n't	O
seem	O
to	O
have	O
the	O
kinds	O
of	O
self-similar	O
structure	O
shown	O
by	O
simple	O
IFS	O
fractals	B-Application
.	O
</s>
<s>
As	O
of	O
1995	O
,	O
all	O
fractal	B-Algorithm
compression	I-Algorithm
software	O
is	O
based	O
on	O
Jacquin	O
's	O
approach	O
.	O
</s>
<s>
The	O
diagram	O
shows	O
the	O
construction	O
on	O
an	O
IFS	O
from	O
two	O
affine	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
The	O
combination	O
of	O
the	O
two	O
functions	O
forms	O
the	O
Hutchinson	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
Three	O
iterations	O
of	O
the	O
operator	O
are	O
shown	O
,	O
and	O
then	O
the	O
final	O
image	O
is	O
of	O
the	O
fixed	O
point	O
,	O
the	O
final	O
fractal	B-Application
.	O
</s>
<s>
Early	O
examples	O
of	O
fractals	B-Application
which	O
may	O
be	O
generated	O
by	O
an	O
IFS	O
include	O
the	O
Cantor	O
set	O
,	O
first	O
described	O
in	O
1884	O
;	O
and	O
de	B-Algorithm
Rham	I-Algorithm
curves	I-Algorithm
,	O
a	O
type	O
of	O
self-similar	O
curve	O
described	O
by	O
Georges	O
de	O
Rham	O
in	O
1957	O
.	O
</s>
<s>
IFSs	O
were	O
conceived	O
in	O
their	O
present	O
form	O
by	O
John	O
E	O
.	O
Hutchinson	O
in	O
1981	O
and	O
popularized	O
by	O
Michael	O
Barnsley	O
's	O
book	O
Fractals	B-Application
Everywhere	O
.	O
</s>
