<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
(	O
)	O
is	O
the	O
set	O
of	O
complex	O
numbers	O
for	O
which	O
the	O
function	O
does	O
not	O
diverge	O
to	O
infinity	B-Application
when	O
iterated	B-Algorithm
from	O
,	O
i.e.	O
,	O
for	O
which	O
the	O
sequence	O
,	O
,	O
etc.	O
,	O
remains	O
bounded	O
in	O
absolute	O
value	O
.	O
</s>
<s>
Images	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
exhibit	O
an	O
elaborate	O
and	O
infinitely	O
complicated	O
boundary	O
that	O
reveals	O
progressively	O
ever-finer	O
recursive	O
detail	O
at	O
increasing	O
magnifications	O
;	O
mathematically	O
,	O
one	O
would	O
say	O
that	O
the	O
boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
a	O
fractal	B-Application
curve	O
.	O
</s>
<s>
Mandelbrot	B-Algorithm
set	I-Algorithm
images	O
may	O
be	O
created	O
by	O
sampling	O
the	O
complex	O
numbers	O
and	O
testing	O
,	O
for	O
each	O
sample	O
point	O
whether	O
the	O
sequence	O
goes	O
to	O
infinity	B-Application
.	O
</s>
<s>
Treating	O
the	O
real	O
and	O
imaginary	O
parts	O
of	O
as	O
image	O
coordinates	O
on	O
the	O
complex	O
plane	O
,	O
pixels	B-Algorithm
may	O
then	O
be	O
coloured	O
according	O
to	O
how	O
soon	O
the	O
sequence	O
crosses	O
an	O
arbitrarily	O
chosen	O
threshold	O
(	O
the	O
threshold	O
has	O
to	O
be	O
at	O
least	O
2	O
,	O
as	O
-2	O
is	O
the	O
complex	O
number	O
with	O
the	O
largest	O
magnitude	O
within	O
the	O
set	O
,	O
but	O
otherwise	O
the	O
threshold	O
is	O
arbitrary	O
)	O
.	O
</s>
<s>
If	O
is	O
held	O
constant	O
and	O
the	O
initial	O
value	O
of	O
is	O
varied	O
instead	O
,	O
one	O
obtains	O
the	O
corresponding	O
Julia	B-Language
set	I-Language
for	O
the	O
point	O
.	O
</s>
<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
has	O
become	O
popular	O
outside	O
mathematics	O
both	O
for	O
its	O
aesthetic	O
appeal	O
and	O
as	O
an	O
example	O
of	O
a	O
complex	O
structure	O
arising	O
from	O
the	O
application	O
of	O
simple	O
rules	O
.	O
</s>
<s>
It	O
is	O
one	O
of	O
the	O
best-known	O
examples	O
of	O
mathematical	B-Application
visualization	I-Application
,	O
mathematical	O
beauty	O
,	O
and	O
motif	O
.	O
</s>
<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
has	O
its	O
origin	O
in	O
complex	O
dynamics	O
,	O
a	O
field	O
first	O
investigated	O
by	O
the	O
French	O
mathematicians	O
Pierre	O
Fatou	O
and	O
Gaston	O
Julia	O
at	O
the	O
beginning	O
of	O
the	O
20th	O
century	O
.	O
</s>
<s>
This	O
fractal	B-Application
was	O
first	O
defined	O
and	O
drawn	O
in	O
1978	O
by	O
Robert	O
W	O
.	O
Brooks	O
and	O
Peter	O
Matelski	O
as	O
part	O
of	O
a	O
study	O
of	O
Kleinian	O
groups	O
.	O
</s>
<s>
Mandelbrot	O
studied	O
the	O
parameter	O
space	O
of	O
quadratic	O
polynomials	O
in	O
an	O
article	B-Algorithm
that	O
appeared	O
in	O
1980	O
.	O
</s>
<s>
The	O
mathematical	O
study	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
really	O
began	O
with	O
work	O
by	O
the	O
mathematicians	O
Adrien	O
Douady	O
and	O
John	O
H	O
.	O
Hubbard	O
(	O
1985	O
)	O
,	O
who	O
established	O
many	O
of	O
its	O
fundamental	O
properties	O
and	O
named	O
the	O
set	O
in	O
honor	O
of	O
Mandelbrot	O
for	O
his	O
influential	O
work	O
in	O
fractal	B-Application
geometry	O
.	O
</s>
<s>
The	O
cover	O
article	B-Algorithm
of	O
the	O
August	O
1985	O
Scientific	O
American	O
introduced	O
a	O
wide	O
audience	O
to	O
the	O
algorithm	O
for	O
computing	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
became	O
prominent	O
in	O
the	O
mid-1980s	O
as	O
a	O
computer	O
graphics	O
demo	O
,	O
when	O
personal	B-Device
computers	I-Device
became	O
powerful	O
enough	O
to	O
plot	O
and	O
display	O
the	O
set	O
in	O
high	O
resolution	O
.	O
</s>
<s>
The	O
work	O
of	O
Douady	O
and	O
Hubbard	O
coincided	O
with	O
a	O
huge	O
increase	O
in	O
interest	O
in	O
complex	O
dynamics	O
and	O
abstract	O
mathematics	O
,	O
and	O
the	O
study	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
has	O
been	O
a	O
centerpiece	O
of	O
this	O
field	O
ever	O
since	O
.	O
</s>
<s>
Thus	O
,	O
a	O
complex	O
number	O
c	O
is	O
a	O
member	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
if	O
,	O
when	O
starting	O
with	O
and	O
applying	O
the	O
iteration	B-Algorithm
repeatedly	O
,	O
the	O
absolute	O
value	O
of	O
remains	O
bounded	O
for	O
all	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
c	O
=	O
1	O
,	O
the	O
sequence	O
is	O
0	O
,	O
1	O
,	O
2	O
,	O
5	O
,	O
26	O
,	O
...	O
,	O
which	O
tends	O
to	O
infinity	B-Application
,	O
so	O
1	O
is	O
not	O
an	O
element	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
can	O
also	O
be	O
defined	O
as	O
the	O
connectedness	O
locus	O
of	O
the	O
family	O
of	O
quadratic	O
polynomials	O
,	O
the	O
subset	O
of	O
the	O
space	O
of	O
parameters	O
for	O
which	O
the	O
Julia	B-Language
set	I-Language
of	O
the	O
corresponding	O
polynomial	O
forms	O
a	O
connected	O
set	O
.	O
</s>
<s>
In	O
the	O
same	O
way	O
,	O
the	O
boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
can	O
be	O
defined	O
as	O
the	O
bifurcation	O
locus	O
of	O
this	O
quadratic	O
family	O
,	O
the	O
subset	O
of	O
parameters	O
near	O
which	O
the	O
dynamic	O
behavior	O
of	O
the	O
polynomial	O
(	O
when	O
it	O
is	O
iterated	B-Algorithm
repeatedly	O
)	O
changes	O
drastically	O
.	O
</s>
<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
a	O
compact	O
set	O
,	O
since	O
it	O
is	O
closed	O
and	O
contained	O
in	O
the	O
closed	O
disk	O
of	O
radius	O
2	O
around	O
the	O
origin	O
.	O
</s>
<s>
More	O
specifically	O
,	O
a	O
point	O
belongs	O
to	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
if	O
and	O
only	O
if	O
for	O
all	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
absolute	O
value	O
of	O
must	O
remain	O
at	O
or	O
below	O
2	O
for	O
to	O
be	O
in	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
,	O
and	O
if	O
that	O
absolute	O
value	O
exceeds	O
2	O
,	O
the	O
sequence	O
will	O
escape	O
to	O
infinity	B-Application
.	O
</s>
<s>
The	O
parameters	O
along	O
this	O
interval	O
can	O
be	O
put	O
in	O
one-to-one	O
correspondence	O
with	O
those	O
of	O
the	O
real	O
logistic	B-Algorithm
family	I-Algorithm
,	O
</s>
<s>
In	O
fact	O
,	O
this	O
gives	O
a	O
correspondence	O
between	O
the	O
entire	O
parameter	O
space	O
of	O
the	O
logistic	B-Algorithm
family	I-Algorithm
and	O
that	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
Douady	O
and	O
Hubbard	O
have	O
shown	O
that	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
connected	O
.	O
</s>
<s>
In	O
fact	O
,	O
they	O
constructed	O
an	O
explicit	O
conformal	O
isomorphism	O
between	O
the	O
complement	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
and	O
the	O
complement	O
of	O
the	O
closed	O
unit	O
disk	O
.	O
</s>
<s>
Mandelbrot	O
had	O
originally	O
conjectured	O
that	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
disconnected	O
.	O
</s>
<s>
There	O
also	O
exists	O
a	O
topological	B-Architecture
proof	O
to	O
the	O
connectedness	O
that	O
was	O
discovered	O
in	O
2001	O
by	O
Jeremy	O
Kahn	O
.	O
</s>
<s>
The	O
dynamical	O
formula	O
for	O
the	O
uniformisation	O
of	O
the	O
complement	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
arising	O
from	O
Douady	O
and	O
Hubbard	O
's	O
proof	O
of	O
the	O
connectedness	O
of	O
,	O
gives	O
rise	O
to	O
external	O
rays	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
These	O
rays	O
can	O
be	O
used	O
to	O
study	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
in	O
combinatorial	O
terms	O
and	O
form	O
the	O
backbone	O
of	O
the	O
Yoccoz	O
parapuzzle	O
.	O
</s>
<s>
As	O
mentioned	O
earlier	O
in	O
the	O
article	B-Algorithm
,	O
the	O
boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
the	O
bifurcation	O
locus	O
of	O
the	O
family	O
of	O
quadratic	O
polynomials	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
the	O
set	O
of	O
all	O
parameters	O
for	O
which	O
the	O
dynamics	O
of	O
the	O
quadratic	O
map	O
exhibits	O
sensitive	O
dependence	O
on	O
i.e.	O
,	O
changes	O
abruptly	O
under	O
arbitrarily	O
small	O
changes	O
of	O
It	O
can	O
be	O
constructed	O
as	O
the	O
limit	O
set	O
of	O
a	O
sequence	O
of	O
plane	O
algebraic	O
curves	O
,	O
the	O
Mandelbrot	O
curves	O
,	O
of	O
the	O
general	O
type	O
known	O
as	O
polynomial	O
lemniscates	O
.	O
</s>
<s>
These	O
algebraic	O
curves	O
appear	O
in	O
images	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
computed	O
using	O
the	O
"	O
escape	O
time	O
algorithm	O
"	O
mentioned	O
below	O
.	O
</s>
<s>
Upon	O
looking	O
at	O
a	O
picture	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
one	O
immediately	O
notices	O
the	O
large	O
cardioid	O
region	O
in	O
the	O
center	O
.	O
</s>
<s>
has	O
an	O
attracting	B-Algorithm
fixed	I-Algorithm
point	I-Algorithm
.	O
</s>
<s>
The	O
reason	O
for	O
the	O
name	O
is	O
that	O
the	O
bulb	O
consists	O
of	O
precisely	O
those	O
parameters	O
for	O
which	O
has	O
an	O
attracting	B-Algorithm
cycle	I-Algorithm
of	I-Algorithm
period	I-Algorithm
2	I-Algorithm
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
periodic	O
Fatou	B-Algorithm
components	I-Algorithm
containing	O
the	O
attracting	O
cycle	O
all	O
touch	O
at	O
a	O
common	O
point	O
(	O
commonly	O
called	O
the	O
-fixed	O
point	O
)	O
.	O
</s>
<s>
The	O
change	O
of	O
behavior	O
occurring	O
at	O
is	O
known	O
as	O
a	O
bifurcation	O
:	O
the	O
attracting	B-Algorithm
fixed	I-Algorithm
point	I-Algorithm
"	O
collides	O
"	O
with	O
a	O
repelling	O
period-q	O
cycle	O
.	O
</s>
<s>
As	O
we	O
pass	O
through	O
the	O
bifurcation	O
parameter	O
into	O
the	O
-bulb	O
,	O
the	O
attracting	B-Algorithm
fixed	I-Algorithm
point	I-Algorithm
turns	O
into	O
a	O
repelling	O
fixed	O
point	O
(	O
the	O
-fixed	O
point	O
)	O
,	O
and	O
the	O
period-q	O
cycle	O
becomes	O
attracting	O
.	O
</s>
<s>
All	O
the	O
bulbs	O
we	O
encountered	O
in	O
the	O
previous	O
section	O
were	O
interior	O
components	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
in	O
which	O
the	O
maps	O
have	O
an	O
attracting	O
periodic	O
cycle	O
.	O
</s>
<s>
Hypothetical	O
non-hyperbolic	O
components	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
are	O
often	O
referred	O
to	O
as	O
"	O
queer	O
"	O
or	O
ghost	O
components	O
.	O
</s>
<s>
Not	O
every	O
hyperbolic	O
component	O
can	O
be	O
reached	O
by	O
a	O
sequence	O
of	O
direct	O
bifurcations	O
from	O
the	O
main	O
cardioid	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
the	O
cycle	O
contains	O
the	O
critical	O
point	O
0	O
,	O
so	O
that	O
0	O
is	O
iterated	B-Algorithm
back	O
to	O
itself	O
after	O
some	O
iterations	B-Algorithm
.	O
</s>
<s>
It	O
is	O
conjectured	O
that	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
locally	O
connected	O
.	O
</s>
<s>
By	O
the	O
work	O
of	O
Adrien	O
Douady	O
and	O
John	O
H	O
.	O
Hubbard	O
,	O
this	O
conjecture	O
would	O
result	O
in	O
a	O
simple	O
abstract	O
"	O
pinched	O
disk	O
"	O
model	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
The	O
work	O
of	O
Jean-Christophe	O
Yoccoz	O
established	O
local	O
connectivity	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
at	O
all	O
finitely	O
renormalizable	O
parameters	O
;	O
that	O
is	O
,	O
roughly	O
speaking	O
those	O
contained	O
only	O
in	O
finitely	O
many	O
small	O
Mandelbrot	O
copies	O
.	O
</s>
<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
self-similar	O
under	O
magnification	O
in	O
the	O
neighborhoods	O
of	O
the	O
Misiurewicz	O
points	O
.	O
</s>
<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
in	O
general	O
is	O
not	O
strictly	O
self-similar	O
but	O
it	O
is	O
quasi-self-similar	O
,	O
as	O
small	O
slightly	O
different	O
versions	O
of	O
itself	O
can	O
be	O
found	O
at	O
arbitrarily	O
small	O
scales	O
.	O
</s>
<s>
These	O
little	O
copies	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
are	O
all	O
slightly	O
different	O
,	O
mostly	O
because	O
of	O
the	O
thin	O
threads	O
connecting	O
them	O
to	O
the	O
main	O
body	O
of	O
the	O
set	O
.	O
</s>
<s>
The	O
Hausdorff	O
dimension	O
of	O
the	O
boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
equals	O
2	O
as	O
determined	O
by	O
a	O
result	O
of	O
Mitsuhiro	O
Shishikura	O
.	O
</s>
<s>
The	O
fact	O
that	O
this	O
is	O
greater	O
(	O
by	O
a	O
whole	O
integer	O
)	O
than	O
its	O
topological	B-Architecture
dimension	O
(	O
which	O
is	O
1	O
)	O
reflects	O
the	O
fractal	B-Application
nature	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
boundary	O
.	O
</s>
<s>
In	O
fact	O
,	O
Shishikura	O
's	O
result	O
intuitively	O
states	O
that	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
boundary	O
is	O
so	O
"	O
wiggly	O
"	O
that	O
it	O
manages	O
to	O
locally	O
fill	O
up	O
space	O
as	O
efficiently	O
as	O
a	O
two-dimensional	O
planar	O
region	O
.	O
</s>
<s>
This	O
opens	O
up	O
the	O
potential	O
possibility	O
that	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
boundary	O
,	O
despite	O
being	O
a	O
curve	O
,	O
has	O
a	O
nonzero	O
area	O
(	O
or	O
,	O
more	O
formally	O
,	O
has	O
a	O
positive	O
planar	O
Lebesgue	O
measure	O
)	O
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
the	O
generalized	O
Mandelbrot	B-Algorithm
set	I-Algorithm
in	O
higher-dimensional	O
hypercomplex	O
number	O
spaces	O
(	O
i.e.	O
</s>
<s>
when	O
the	O
power	O
of	O
the	O
iterated	B-Algorithm
variable	O
tends	O
to	O
infinity	B-Application
)	O
is	O
convergent	O
to	O
the	O
unit	O
(	O
-1	O
)	O
-sphere	O
.	O
</s>
<s>
In	O
the	O
Blum	B-Application
–	I-Application
Shub	I-Application
–	I-Application
Smale	I-Application
model	O
of	O
real	O
computation	O
,	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
not	O
computable	O
,	O
but	O
its	O
complement	O
is	O
computably	O
enumerable	O
.	O
</s>
<s>
At	O
present	O
,	O
it	O
is	O
unknown	O
whether	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
computable	O
in	O
models	O
of	O
real	O
computation	O
based	O
on	O
computable	O
analysis	O
,	O
which	O
correspond	O
more	O
closely	O
to	O
the	O
intuitive	O
notion	O
of	O
"	O
plotting	O
the	O
set	O
by	O
a	O
computer	O
"	O
.	O
</s>
<s>
Hertling	O
has	O
shown	O
that	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
computable	O
in	O
this	O
model	O
if	O
the	O
hyperbolicity	O
conjecture	O
is	O
true	O
.	O
</s>
<s>
As	O
a	O
consequence	O
of	O
the	O
definition	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
there	O
is	O
a	O
close	O
correspondence	O
between	O
the	O
geometry	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
at	O
a	O
given	O
point	O
and	O
the	O
structure	O
of	O
the	O
corresponding	O
Julia	B-Language
set	I-Language
.	O
</s>
<s>
For	O
instance	O
,	O
a	O
value	O
of	O
c	O
belongs	O
to	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
if	O
the	O
corresponding	O
Julia	B-Language
set	I-Language
is	O
connected	O
.	O
</s>
<s>
Thus	O
,	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
may	O
be	O
seen	O
as	O
a	O
map	O
of	O
the	O
connected	O
Julia	B-Language
sets	I-Language
.	O
</s>
<s>
This	O
principle	O
is	O
exploited	O
in	O
virtually	O
all	O
deep	O
results	O
on	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
Shishikura	O
proved	O
that	O
,	O
for	O
a	O
dense	O
set	O
of	O
parameters	O
in	O
the	O
boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
the	O
Julia	B-Language
set	I-Language
has	O
Hausdorff	O
dimension	O
two	O
,	O
and	O
then	O
transfers	O
this	O
information	O
to	O
the	O
parameter	O
plane	O
.	O
</s>
<s>
Similarly	O
,	O
Yoccoz	O
first	O
proved	O
the	O
local	O
connectivity	O
of	O
Julia	B-Language
sets	I-Language
,	O
before	O
establishing	O
it	O
for	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
at	O
the	O
corresponding	O
parameters	O
.	O
</s>
<s>
The	O
part	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
connected	O
to	O
the	O
main	O
cardioid	O
at	O
this	O
bifurcation	O
point	O
is	O
called	O
the	O
p/q	O
-limb	O
.	O
</s>
<s>
In	O
an	O
attempt	O
to	O
demonstrate	O
that	O
the	O
thickness	O
of	O
the	O
p/q	O
-limb	O
is	O
zero	O
,	O
David	O
Boll	O
carried	O
out	O
a	O
computer	O
experiment	O
in	O
1991	O
,	O
where	O
he	O
computed	O
the	O
number	O
of	O
iterations	B-Algorithm
required	O
for	O
the	O
series	O
to	O
diverge	O
for	O
(	O
being	O
the	O
location	O
thereof	O
)	O
.	O
</s>
<s>
As	O
the	O
series	O
does	O
not	O
diverge	O
for	O
the	O
exact	O
value	O
of	O
,	O
the	O
number	O
of	O
iterations	B-Algorithm
required	O
increases	O
with	O
a	O
small	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
multiplying	O
the	O
value	O
of	O
with	O
the	O
number	O
of	O
iterations	B-Algorithm
required	O
yields	O
an	O
approximation	O
of	O
that	O
becomes	O
better	O
for	O
smaller	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
=	O
0.0000001	O
the	O
number	O
of	O
iterations	B-Algorithm
is	O
31415928	O
and	O
the	O
product	O
is	O
3.1415928	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
is	O
located	O
within	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
and	O
that	O
a	O
relation	O
exists	O
between	O
the	O
main	O
cardioid	O
and	O
the	O
Farey	O
Diagram	O
.	O
</s>
<s>
Upon	O
mapping	O
the	O
main	O
cardioid	O
to	O
a	O
disk	O
,	O
one	O
can	O
notice	O
that	O
the	O
amount	O
of	O
antennae	O
that	O
extends	O
from	O
the	O
next	O
largest	O
Hyperbolic	O
component	O
,	O
and	O
that	O
is	O
located	O
between	O
the	O
two	O
previously	O
selected	O
components	O
,	O
follows	O
suit	O
with	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
of	O
1	O
,	O
2	O
,	O
3	O
,	O
5	O
,	O
8	O
,	O
13	O
,	O
and	O
21	O
can	O
be	O
found	O
within	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
The	O
boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
shows	O
more	O
intricate	O
detail	O
the	O
closer	O
one	O
looks	O
or	O
magnifies	O
the	O
image	O
,	O
usually	O
called	O
"	O
zooming	O
in	O
"	O
.	O
</s>
<s>
The	O
following	O
example	O
of	O
an	O
image	O
sequence	O
zooming	O
to	O
a	O
selected	O
c	O
value	O
gives	O
an	O
impression	O
of	O
the	O
infinite	O
richness	O
of	O
different	O
geometrical	O
structures	O
present	O
in	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
boundary	O
and	O
explains	O
some	O
of	O
their	O
typical	O
rules	O
.	O
</s>
<s>
Relating	O
to	O
an	O
ordinary	O
computer	B-Device
monitor	I-Device
,	O
it	O
represents	O
a	O
section	O
of	O
a	O
Mandelbrot	B-Algorithm
set	I-Algorithm
with	O
a	O
diameter	O
of	O
4	O
million	O
kilometers	O
.	O
</s>
<s>
Its	O
border	O
shows	O
an	O
astronomical	O
number	O
of	O
different	O
fractal	B-Application
structures	O
.	O
</s>
<s>
These	O
two	O
groups	O
can	O
be	O
attributed	O
by	O
some	O
kind	O
of	O
metamorphosis	O
to	O
the	O
two	O
"	O
fingers	O
"	O
of	O
the	O
"	O
upper	O
hand	O
"	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
;	O
therefore	O
,	O
the	O
number	O
of	O
"	O
spokes	O
"	O
increases	O
from	O
one	O
"	O
seahorse	O
"	O
to	O
the	O
next	O
by	O
2	O
;	O
the	O
"	O
hub	O
"	O
is	O
a	O
so-called	O
Misiurewicz	O
point	O
.	O
</s>
<s>
Between	O
the	O
"	O
upper	O
part	O
of	O
the	O
body	O
"	O
and	O
the	O
"	O
tail	O
"	O
a	O
distorted	O
small	O
copy	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
called	O
satellite	O
may	O
be	O
recognized	O
.	O
</s>
<s>
The	O
islands	O
in	O
the	O
third-to-last	O
step	O
seem	O
to	O
consist	O
of	O
infinitely	O
many	O
parts	O
like	O
Cantor	O
sets	O
,	O
as	O
is	O
actually	O
the	O
case	O
for	O
the	O
corresponding	O
Julia	B-Language
set	I-Language
.	O
</s>
<s>
The	O
value	O
of	O
for	O
the	O
corresponding	O
is	O
not	O
that	O
of	O
the	O
image	O
center	O
but	O
,	O
relative	O
to	O
the	O
main	O
body	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
has	O
the	O
same	O
position	O
as	O
the	O
center	O
of	O
this	O
image	O
relative	O
to	O
the	O
satellite	O
shown	O
in	O
the	O
6th	O
zoom	O
step	O
.	O
</s>
<s>
While	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
typically	O
rendered	O
showing	O
the	O
outside	O
boundary	O
detail	O
,	O
structure	O
within	O
the	O
bounded	O
set	O
can	O
also	O
be	O
revealed	O
.	O
</s>
<s>
For	O
an	O
integer	O
d	O
,	O
these	O
sets	O
are	O
connectedness	O
loci	O
for	O
the	O
Julia	B-Language
sets	I-Language
built	O
from	O
the	O
same	O
formula	O
.	O
</s>
<s>
For	O
general	O
families	O
of	O
holomorphic	O
functions	O
,	O
the	O
boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
generalizes	O
to	O
the	O
bifurcation	O
locus	O
,	O
which	O
is	O
a	O
natural	O
object	O
to	O
study	O
even	O
when	O
the	O
connectedness	O
locus	O
is	O
not	O
useful	O
.	O
</s>
<s>
The	O
Multibrot	B-Algorithm
set	I-Algorithm
is	O
obtained	O
by	O
varying	O
the	O
value	O
of	O
the	O
exponent	O
d	O
.	O
The	O
article	B-Algorithm
has	O
a	O
video	O
that	O
shows	O
the	O
development	O
from	O
d	O
=	O
0	O
to	O
7	O
,	O
at	O
which	O
point	O
there	O
are	O
6	O
i.e.	O
</s>
<s>
lobes	O
around	O
the	O
perimeter	B-Application
.	O
</s>
<s>
There	O
is	O
no	O
perfect	O
extension	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
into	O
3D	O
.	O
</s>
<s>
This	O
is	O
because	O
there	O
is	O
no	O
3D	O
analogue	O
of	O
the	O
complex	O
numbers	O
for	O
it	O
to	O
iterate	B-Algorithm
on	O
.	O
</s>
<s>
However	O
,	O
there	O
is	O
an	O
extension	O
of	O
the	O
complex	O
numbers	O
into	O
4	O
dimensions	O
,	O
called	O
the	O
quaternions	O
,	O
that	O
creates	O
a	O
perfect	O
extension	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
and	O
the	O
Julia	B-Language
sets	I-Language
into	O
4	O
dimensions	O
.	O
</s>
<s>
These	O
can	O
then	O
be	O
either	O
cross-sectioned	B-Application
or	O
projected	B-General_Concept
into	O
a	O
3D	O
structure	O
.	O
</s>
<s>
However	O
,	O
the	O
quaternion	O
(	O
4-dimensional	O
)	O
Mandelbrot	B-Algorithm
set	I-Algorithm
is	O
simply	O
a	O
solid	O
of	O
revolution	O
of	O
the	O
2-dimensional	O
Mandelbrot	B-Algorithm
set	I-Algorithm
(	O
in	O
the	O
j-k	O
plane	O
)	O
,	O
and	O
is	O
therefore	O
largely	O
uninteresting	O
to	O
look	O
at	O
,	O
as	O
it	O
does	O
not	O
have	O
any	O
of	O
the	O
properties	O
that	O
would	O
be	O
expected	O
from	O
a	O
4-dimensional	O
fractal	B-Application
.	O
</s>
<s>
Taking	O
a	O
3-dimensional	O
cross	B-Application
section	I-Application
at	O
results	O
in	O
a	O
solid	O
of	O
revolution	O
of	O
the	O
2-dimensional	O
Mandelbrot	B-Algorithm
set	I-Algorithm
around	O
the	O
real	O
axis	O
.	O
</s>
<s>
The	O
tricorn	B-Algorithm
(	O
also	O
sometimes	O
called	O
the	O
Mandelbar	B-Algorithm
)	O
was	O
encountered	O
by	O
Milnor	O
in	O
his	O
study	O
of	O
parameter	O
slices	O
of	O
real	O
cubic	O
polynomials	O
.	O
</s>
<s>
Another	O
non-analytic	O
generalization	O
is	O
the	O
Burning	B-Algorithm
Ship	I-Algorithm
fractal	I-Algorithm
,	O
which	O
is	O
obtained	O
by	O
iterating	O
the	O
following	O
:	O
</s>
<s>
There	O
exist	O
a	O
multitude	O
of	O
various	O
algorithms	O
for	O
plotting	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
via	O
a	O
computing	O
device	O
.	O
</s>
<s>
In	O
the	O
escape	O
time	O
algorithm	O
,	O
a	O
repeating	O
calculation	O
is	O
performed	O
for	O
each	O
x	O
,	O
y	O
point	O
in	O
the	O
plot	O
area	O
and	O
based	O
on	O
the	O
behavior	O
of	O
that	O
calculation	O
,	O
a	O
color	O
is	O
chosen	O
for	O
that	O
pixel	B-Algorithm
.	O
</s>
<s>
The	O
result	O
of	O
each	O
iteration	B-Algorithm
is	O
used	O
as	O
the	O
starting	O
values	O
for	O
the	O
next	O
.	O
</s>
<s>
The	O
values	O
are	O
checked	O
during	O
each	O
iteration	B-Algorithm
to	O
see	O
whether	O
they	O
have	O
reached	O
a	O
critical	O
"	O
escape	O
"	O
condition	O
,	O
or	O
"	O
bailout	O
"	O
.	O
</s>
<s>
If	O
that	O
condition	O
is	O
reached	O
,	O
the	O
calculation	O
is	O
stopped	O
,	O
the	O
pixel	B-Algorithm
is	O
drawn	O
,	O
and	O
the	O
next	O
x	O
,	O
y	O
point	O
is	O
examined	O
.	O
</s>
<s>
Often	O
black	O
is	O
used	O
to	O
show	O
values	O
that	O
fail	O
to	O
escape	O
before	O
the	O
iteration	B-Algorithm
limit	O
,	O
and	O
gradually	O
brighter	O
colors	O
are	O
used	O
for	O
points	O
that	O
escape	O
.	O
</s>
<s>
To	O
render	O
such	O
an	O
image	O
,	O
the	O
region	O
of	O
the	O
complex	O
plane	O
we	O
are	O
considering	O
is	O
subdivided	O
into	O
a	O
certain	O
number	O
of	O
pixels	B-Algorithm
.	O
</s>
<s>
To	O
color	O
any	O
such	O
pixel	B-Algorithm
,	O
let	O
be	O
the	O
midpoint	O
of	O
that	O
pixel	B-Algorithm
.	O
</s>
<s>
We	O
now	O
iterate	B-Algorithm
the	O
critical	O
point	O
0	O
under	O
,	O
checking	O
at	O
each	O
step	O
whether	O
the	O
orbit	B-Algorithm
point	O
has	O
a	O
radius	O
larger	O
than	O
2	O
.	O
</s>
<s>
When	O
this	O
is	O
the	O
case	O
,	O
we	O
know	O
that	O
does	O
not	O
belong	O
to	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
and	O
we	O
color	O
our	O
pixel	B-Algorithm
according	O
to	O
the	O
number	O
of	O
iterations	B-Algorithm
used	O
to	O
find	O
out	O
.	O
</s>
<s>
Otherwise	O
,	O
we	O
keep	O
iterating	O
up	O
to	O
a	O
fixed	O
number	O
of	O
steps	O
,	O
after	O
which	O
we	O
decide	O
that	O
our	O
parameter	O
is	O
"	O
probably	O
"	O
in	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
or	O
at	O
least	O
very	O
close	O
to	O
it	O
,	O
and	O
color	O
the	O
pixel	B-Algorithm
black	O
.	O
</s>
<s>
In	O
pseudocode	B-Language
,	O
this	O
algorithm	O
would	O
look	O
as	O
follows	O
.	O
</s>
<s>
Here	O
,	O
relating	O
the	O
pseudocode	B-Language
to	O
,	O
and	O
:	O
</s>
<s>
and	O
so	O
,	O
as	O
can	O
be	O
seen	O
in	O
the	O
pseudocode	B-Language
in	O
the	O
computation	O
of	O
x	O
and	O
y	O
:	O
</s>
<s>
To	O
get	O
colorful	O
images	O
of	O
the	O
set	O
,	O
the	O
assignment	O
of	O
a	O
color	O
to	O
each	O
value	O
of	O
the	O
number	O
of	O
executed	O
iterations	B-Algorithm
can	O
be	O
made	O
using	O
one	O
of	O
a	O
variety	O
of	O
functions	O
(	O
linear	O
,	O
exponential	O
,	O
etc	O
.	O
</s>
<s>
The	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
is	O
widely	O
considered	O
the	O
most	O
popular	O
fractal	B-Application
,	O
and	O
has	O
been	O
referenced	O
several	O
times	O
in	O
popular	O
culture	O
.	O
</s>
<s>
The	O
Jonathan	O
Coulton	O
song	O
"	O
Mandelbrot	B-Algorithm
Set	I-Algorithm
"	O
is	O
a	O
tribute	O
to	O
both	O
the	O
fractal	B-Application
itself	O
and	O
to	O
the	O
man	O
it	O
is	O
named	O
after	O
,	O
Benoit	O
Mandelbrot	O
.	O
</s>
<s>
The	O
second	O
book	O
of	O
the	O
Mode	O
series	O
by	O
Piers	O
Anthony	O
,	O
Fractal	B-Application
Mode	O
,	O
describes	O
a	O
world	O
that	O
is	O
a	O
perfect	O
3D	O
model	O
of	O
the	O
set	O
.	O
</s>
<s>
The	O
Arthur	O
C	O
.	O
Clarke	O
novel	O
The	O
Ghost	O
from	O
the	O
Grand	O
Banks	O
features	O
an	O
artificial	O
lake	O
made	O
to	O
replicate	O
the	O
shape	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
The	O
American	O
rock	O
band	O
Heart	O
has	O
an	O
image	O
of	O
a	O
Mandelbrot	B-Algorithm
set	I-Algorithm
on	O
the	O
cover	O
of	O
their	O
2004	O
album	O
,	O
Jupiters	O
Darling	O
.	O
</s>
<s>
The	O
British	O
black	O
metal	O
band	O
Anaal	O
Nathrakh	O
uses	O
an	O
image	O
resembling	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
on	O
their	O
Eschaton	O
album	O
cover	O
art	O
.	O
</s>
<s>
The	O
television	O
series	O
Dirk	O
Gently	O
's	O
Holistic	O
Detective	O
Agency	O
(	O
2016	O
)	O
prominently	O
features	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
in	O
connection	O
with	O
the	O
visions	O
of	O
the	O
character	O
Amanda	O
.	O
</s>
<s>
In	O
the	O
second	O
season	O
,	O
her	O
jacket	O
has	O
a	O
large	O
image	O
of	O
the	O
fractal	B-Application
on	O
the	O
back	O
.	O
</s>
<s>
In	O
Ian	O
Stewart	O
's	O
2001	O
book	O
Flatterland	O
,	O
there	O
is	O
a	O
character	O
called	O
the	O
Mandelblot	O
,	O
who	O
helps	O
explain	O
fractals	B-Application
to	O
the	O
characters	O
and	O
reader	O
.	O
</s>
