<s>
The	O
Buddhabrot	B-Algorithm
is	O
the	O
probability	O
distribution	O
over	O
the	O
trajectories	O
of	O
points	O
that	O
escape	O
the	O
Mandelbrot	B-Algorithm
fractal	I-Algorithm
.	O
</s>
<s>
The	O
Buddhabrot	B-Algorithm
rendering	O
technique	O
was	O
discovered	O
by	O
Melinda	B-Algorithm
Green	I-Algorithm
,	O
who	O
later	O
described	O
it	O
in	O
a	O
1993	O
Usenet	B-Application
post	O
to	O
sci.fractals	O
.	O
</s>
<s>
Previous	O
researchers	O
had	O
come	O
very	O
close	O
to	O
finding	O
the	O
precise	O
Buddhabrot	B-Algorithm
technique	O
.	O
</s>
<s>
This	O
led	O
directly	O
to	O
the	O
discovery	O
of	O
Pickover	B-Algorithm
stalks	I-Algorithm
.	O
</s>
<s>
Noel	O
Griffin	O
also	O
implemented	O
this	O
idea	O
in	O
the	O
1993	O
"	O
Mandelcloud	O
"	O
option	O
in	O
the	O
Fractint	B-Application
renderer	O
.	O
</s>
<s>
The	O
inverse	O
,	O
"	O
Anti-Buddhabrot	O
"	O
filter	O
produces	O
images	O
similar	O
to	O
no	O
filtering	O
.	O
</s>
<s>
The	O
name	O
Buddhabrot	B-Algorithm
was	O
coined	O
later	O
by	O
Lori	O
Gardi	O
.	O
</s>
<s>
does	O
tend	O
to	O
infinity	B-Application
as	O
goes	O
to	O
infinity	B-Application
for	O
.	O
</s>
<s>
The	O
Buddhabrot	B-Algorithm
image	O
can	O
be	O
constructed	O
by	O
first	O
creating	O
a	O
2-dimensional	O
array	B-Data_Structure
of	O
boxes	O
,	O
each	O
corresponding	O
to	O
a	O
final	O
pixel	O
in	O
the	O
image	O
.	O
</s>
<s>
Next	O
,	O
a	O
random	O
sampling	O
of	O
points	O
are	O
iterated	B-Algorithm
through	O
the	O
Mandelbrot	O
function	O
.	O
</s>
<s>
For	O
points	O
which	O
escape	O
within	O
a	O
chosen	O
maximum	O
number	O
of	O
iterations	B-Algorithm
,	O
and	O
therefore	O
are	O
not	O
in	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
the	O
counter	O
for	O
each	O
box	O
entered	O
during	O
the	O
escape	O
to	O
infinity	B-Application
is	O
incremented	O
by	O
1	O
.	O
</s>
<s>
Points	O
which	O
do	O
not	O
escape	O
within	O
the	O
maximum	O
number	O
of	O
iterations	B-Algorithm
(	O
and	O
considered	O
to	O
be	O
in	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
)	O
are	O
discarded	O
.	O
</s>
<s>
After	O
a	O
large	O
number	O
of	O
values	O
have	O
been	O
iterated	B-Algorithm
,	O
grayscale	O
shades	O
are	O
then	O
chosen	O
based	O
on	O
the	O
distribution	O
of	O
values	O
recorded	O
in	O
the	O
array	B-Data_Structure
.	O
</s>
<s>
The	O
result	O
is	O
a	O
density	O
plot	O
highlighting	O
regions	O
where	O
values	O
spend	O
the	O
most	O
time	O
on	O
their	O
way	O
to	O
infinity	B-Application
.	O
</s>
<s>
Rendering	O
Buddhabrot	B-Algorithm
images	O
is	O
typically	O
more	O
computationally	O
intensive	O
than	O
standard	O
Mandelbrot	O
rendering	O
techniques	O
.	O
</s>
<s>
This	O
is	O
partly	O
due	O
to	O
requiring	O
more	O
random	O
points	O
to	O
be	O
iterated	B-Algorithm
than	O
pixels	O
in	O
the	O
image	O
in	O
order	O
to	O
build	O
up	O
a	O
sharp	O
image	O
.	O
</s>
<s>
Conversely	O
,	O
a	O
pixel	O
in	O
a	O
zoomed	O
region	O
of	O
a	O
Buddhabrot	B-Algorithm
image	O
can	O
be	O
affected	O
by	O
initial	O
points	O
from	O
regions	O
far	O
outside	O
the	O
one	O
being	O
rendered	O
.	O
</s>
<s>
Without	O
resorting	O
to	O
more	O
complex	O
probabilistic	O
techniques	O
,	O
rendering	O
zoomed	O
portions	O
of	O
Buddhabrot	B-Algorithm
consists	O
of	O
merely	O
cropping	O
a	O
large	O
full	O
sized	O
rendering	O
.	O
</s>
<s>
The	O
maximum	O
number	O
of	O
iterations	B-Algorithm
chosen	O
affects	O
the	O
image	O
–	O
higher	O
values	O
give	O
sparser	O
more	O
detailed	O
appearance	O
,	O
as	O
a	O
few	O
of	O
the	O
points	O
pass	O
through	O
a	O
large	O
number	O
of	O
pixels	O
before	O
they	O
escape	O
,	O
resulting	O
in	O
their	O
paths	O
being	O
more	O
prominent	O
.	O
</s>
<s>
Some	O
of	O
these	O
streaks	O
are	O
visible	O
in	O
the	O
1,000,000	O
iteration	B-Algorithm
image	O
below	O
.	O
</s>
<s>
Green	O
later	O
realized	O
that	O
this	O
provided	O
a	O
natural	O
way	O
to	O
create	O
color	O
Buddhabrot	B-Algorithm
images	O
by	O
taking	O
three	O
such	O
grayscale	O
images	O
,	O
differing	O
only	O
by	O
the	O
maximum	O
number	O
of	O
iterations	B-Algorithm
used	O
,	O
and	O
combining	O
them	O
into	O
a	O
single	O
color	O
image	O
using	O
the	O
same	O
method	O
used	O
by	O
astronomers	O
to	O
create	O
false	B-Application
color	I-Application
images	I-Application
of	O
nebula	O
and	O
other	O
celestial	O
objects	O
.	O
</s>
<s>
For	O
example	O
,	O
one	O
could	O
assign	O
a	O
2,000	O
max	O
iteration	B-Algorithm
image	O
to	O
the	O
red	O
channel	O
,	O
a	O
200	O
max	O
iteration	B-Algorithm
image	O
to	O
the	O
green	O
channel	O
,	O
and	O
a	O
20	O
max	O
iteration	B-Algorithm
image	O
to	O
the	O
blue	O
channel	O
of	O
an	O
image	O
in	O
an	O
RGB	O
color	O
space	O
.	O
</s>
<s>
Some	O
have	O
labelled	O
Buddhabrot	B-Algorithm
images	O
using	O
this	O
technique	O
Nebulabrots	B-Algorithm
.	O
</s>
<s>
The	O
relationship	O
between	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
as	O
defined	O
by	O
the	O
iteration	B-Algorithm
,	O
and	O
the	O
logistic	B-Algorithm
map	I-Algorithm
is	O
well	O
known	O
.	O
</s>
<s>
The	O
traditional	O
way	O
of	O
illustrating	O
this	O
relationship	O
is	O
aligning	O
the	O
logistic	B-Algorithm
map	I-Algorithm
and	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
through	O
the	O
relation	O
between	O
and	O
,	O
using	O
a	O
common	O
x-axis	O
and	O
a	O
different	O
y-axis	O
,	O
showing	O
a	O
one-dimensional	O
relationship	O
.	O
</s>
<s>
Melinda	B-Algorithm
Green	I-Algorithm
discovered	O
'	O
by	O
accident	O
 '	O
that	O
the	O
Anti-Buddhabrot	O
paradigm	O
fully	O
integrates	O
the	O
logistic	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Both	O
are	O
based	O
on	O
tracing	O
paths	O
from	O
non-escaping	O
points	O
,	O
iterated	B-Algorithm
from	O
a	O
(	O
random	O
)	O
starting	O
point	O
,	O
and	O
the	O
iteration	B-Algorithm
functions	O
are	O
related	O
by	O
the	O
transformation	O
given	O
above	O
.	O
</s>
<s>
It	O
is	O
then	O
easy	O
to	O
see	O
that	O
the	O
Anti-Buddhabrot	O
for	O
,	O
plotting	O
paths	O
with	O
and	O
,	O
simply	O
generates	O
the	O
logistic	B-Algorithm
map	I-Algorithm
in	O
the	O
plane	O
,	O
when	O
using	O
the	O
given	O
transformation	O
.	O
</s>
<s>
In	O
the	O
logistic	B-Algorithm
map	I-Algorithm
,	O
all	O
ultimately	O
generate	O
the	O
same	O
path	O
.	O
</s>
<s>
,	O
and	O
also	O
the	O
Anti-Buddhabrot	O
with	O
and	O
,	O
this	O
is	O
the	O
2D	O
logistic	B-Algorithm
map	I-Algorithm
in	O
the	O
plane	O
.	O
</s>
<s>
The	O
logistic	B-Algorithm
map	I-Algorithm
Anti-Buddhabrot	O
is	O
in	O
fact	O
a	O
subset	O
of	O
the	O
classic	O
Anti-Buddhabrot	O
,	O
situated	O
in	O
the	O
plane	O
(	O
or	O
)	O
of	O
3D	O
,	O
perpendicular	O
to	O
the	O
plane	O
.	O
</s>
